# Isoparametric elements and numerical integration mcq

# Isoparametric elements and numerical integration mcq

isoparametric elements and numerical integration mcq T d q. However for nonlinear equa tions of motion the use of numerical integration procedures appears to be mandatory. Reissner theory sub element s is Numerical integration in the finite element method Computers amp Structures Vol. Keywords numerical integration level set un tted nite element method isoparametric nite element method high order methods interface problems 1. 257 289. 2 to 2. Earlier implementation of the method used the same Gauss integration scheme as for the integration of element stiffness matrix 10 . The new set of quadrature consist of 13 sampling points and weights out which 12 points are at the edges of the brick element and one point is considered at the center of the element. It allowed very accurate higher order elements of arbitrary shape to be developed and programmed with a minimum of effort. Isoparametric representations prevent closed form integration over individual elements and numerical integration is required. Motivation Finite Element Formulation for Two Dimensional Scalar Field Problems Finite Element Discretization and Piecewise Approximation Three Node Triangular Finite Element Four Node Rectangular Element Isoparametric Finite Elements and the Four Node Quadrilateral 4Q Element Numerical Integration for Isoparametric Quadrilateral Elements Higher Order Another important step in finite element analysis is the integration over the elements. Equilibrium equations in elasticity subjected to body force traction forces stress strain relations for plane stress and plane strain Boundary conditions Initial conditions Euler s Lagrange s equations of bar beams Principal of a minimum potential energy principle of virtual work Rayleigh Ritz method Galerkins method. Stiffness of Truss FEA Quiz worksheets 1. 3 6 node isoparametric triangul r element T6 26 2. It is possible thus to use fewer elements for the solution. The matrix should be symmetric and for a symmetric positive The enrichment function is used to improve the shape function of classical finite element method FEM for the nodes supporting the elements cut by the interface. Multiple Choice Questions Truss element 12 Equation 22 can be integrated analytically here. Taking the derivative of this expression to find the Jacobian terms The isoparametric element Rao 1982 is a powerful tool that uses local coordinates to perform the integration over the element. This may be carried out before or together with discretization using either a plate or shell theory or directly the three dimensional field equations. The construction of hexahedra shape functions and the computation of the stiffness matrix was greatly facilitated by three advances in finite element technology natural coordinates isoparametric description and numerical integration 25 . A theoretical explanation is given for the observed loss of accuracy in highly distorted isoparametric elements. 1 Introduction 242 14. Within those integration elements the integrand is smooth and hence numerical integration can be performed properly. Rathod 21 presented analytical integration formulas for a four node isoparametric finite element and showed that all the integration formulas could be obtained on the basis of four simple integrals. through the Biot Savart Equation Using Curvilinear Elements andanAdaptiveProcess Anderson Nunes1 2 Olivier Chadebec3 Patrick Kuo Peng 1 Patrick Dular 4 and Bruno Cucco5 Abstract The analytical solution of the Biot Savart equation can be complex in some cases and its numerical integration is commonly more appropriate. Keywords Isoparametric Interpolating functions Gauss Quadrature Numerical Integration etc. Master numerical methods of linear algebraic equations. Meshing using A presentation of detailed theory and computer programs which can be used for stress analysis. Variational equations Galerkin s method. 9 projects 7. What is the value of x at 0and what is the value of the Jacobian there 5. 3 6 0 J 2 0 3 7. The basis of the isoparametric formulation for 2D solid elements is described in the next section. 3Cubic Elements 293 5. quot The interpolating functions are shown below 4 ISOPARAMETRIC MEMBRANE AND SOLID ELEMENTS 4. 6. in each direction are used Plane stress Rectangular element Plane stress Triangular elements Natural coordinates and Isoparametric elements Quadrilateral isoparametric elements Numerical integration Steady state field problems Plate bending elements 2 D creeping flow General procedures for solving nonlinear discrete equations Applications Numerical Integration 161 8. Such an interpolation yields a general relation ship between the natural and cartesian coordinates. Quad8 computed by symbolic integration. 14 Elemental Form and Layout 173 8. 2 7. 8 9 Isoparametric elements triangular quadrilateral tetrahedral hexahedral Element sti ness assembly operations FEM 3. 38 is a four node three side isoparametric solid tetrahedron element. 16 Patch Test 179. 4 numerical integration 7. 8 Statics condensation effects of underintegration. This region was improved Reissner for Mindlin Theory. Problems Numerical Integration. Numerical integration 2. An unfitted finite element method which is suitable for the case of piecewise planar interfaces is combined with a parametric mapping of the underlying mesh resulting in an isoparametric unfitted 7. No Question Answer 1 In which of the methods of engineering analysis prototype are used Experimental 2 The finite difference method is used for solving_____ kind of problems Heat Transfer amp structural 3 The To better understand the behavior of Gaussian quadrature integration the isoparametric element is formulated with three types of numerical integration 1 the two by two Gaussian quadrature rule 2 the three by three Gaussian quadrature rule and 3 a fully converged numerical integration typically using 400 subdivisions . 2 PARTIALDERIVATIVECOMPUTATION 17. We introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions. The techniques described in the previous sections allow meshing of irregular domains with compatible elements i. The arbitrary shape and general course of the loaded area in nodal points is allowed by use of 4 and 8 node isoparametric elements numerical integration and Jacobean transformation. of the finite element adopted an infinite element method. On exact integration within an isoparametric tetragonal Finite Element International Conference of Numerical Analysis and Applied Mathematics AIP Conference Proceedings 936 582 585. Numerical integration Exercise 1 For the 3 noded bar element 1 Derive the expression of the deformation matrix in isoparametric coordinates B r dN r dr dr dx 2 Determine the required order for the sti ness matrix integration 3 Compare the numerically evaluated sti ness matrix with the analytical one K Z 1 1 BTEAB detJ dr Institute of Stiffness matrix K B T D B 2 p rA 66. Numerical integration of equations of motions algorithms and their numerical stability simple examples. The local average method of a three dimensional irregu lar element is adopted and the infinite element is a non isoparametric element which makes the analysis results method element shapes finite element analysis finite element analysis book by jalaluddin md pdf pea fea beam elements fea two dimessional problem lagrangian serenalipity elements isoparametric formulation numerical integration etc. 8 Numerical Examples 232 Questions 240 References 241 14. 3 pp. minglassing B. Make it possible to choose a variety of factors. Two Dimensional Problems Second order single variable equations Isoparametric elements and numerical integration Second order multivariable equations Computational strategies 5. 17 3 17. Assemble the basic finite difference and finite element equations. 1. For a rectangular element 3x3 Gauss integration gives exact K matrix. NO E61. 4 A Three Dimensional Rigid Frame Element 266 14. The side numbering is following. 6 isoparametric triangular elements 7. In Introductory Lectures on the Finite Element Method. Finite element method practical aspects a utomatic mesh generation adaptive techniques. Barlow did not specifically use the term Features of a finite element code Introduction to ABAQUS CAE and ABAQUS 2. Introduction 1. Curved isoparametric quadriltaeral elements for finite element analysis 37 As Ni is defined in terms of 5 and q it is necessary to change the derivatives to a ax and a . Element3D Base class for all 3D FEM elements. Usually the isoparametric based boundary element uses same polynomial order in the both basis function and mapping function. In each integration point the integrand is calculated and the result is multiplied by a weighting factor and added to the existing element value. 7Examples of One Dimensional Elements in ANSYS 305 Summary 305 References 305 Problems 305 Practical advice Assuming an isoparametric formulation in the UEL you use quot element shape functions quot and the primary field variables defined by the nodal degrees of freedom with a solid mechanics focus these are typically the displacements to calculate the element strains stresses etc. There is given a detailed analysis of a super convergence phenomenon for the gradient of approximate solutions to second order Numerical illustrations for a con Stiffness Matrices of Isoparametric Four Node Finite Elements by Exact Analytical Integration Journal of Aerospace Engineering Vol 21 No 2 Sponsored by the Aerospace Division 7. The components of displacements parallel to x y axis at the modes 1 2 3 are found to be In your finite element models you may encounter the concept of numerical integration and Gauss points in several contexts. Figure 3 6 Transformation of a finite element graph consisting of rectangles to a square unit element master element taken from Reddy et al. Similar to the case of two dimensional elements one can produce more accurate and useful elements by using fewer points. The use of full Gaussian quadrature second order requires 2 2 integration points in two dimensions e. 5 Section 5. Q 9 Where denotes the 2. 2 POLYNOMIAL FORM OF INTERPOLATION FUNCTIONS If a polynomial type of variation is assumed for the field variable x ina one dimensional element x can be element method. Introduction In this Chapter the isoparametric representation of element geometry and shape functions discussed in the previous Chapter is used to construct quadrilateral elements for the plane stress problem. 1972 Isoparametric elements and numerical integration. At left the integration point is located at the barycenter of Finite elements III Local and global coordinate systems. Isoparametric Mapping in the Two Dimensional Case of a 4 node Bilinear Element The requirement that the element area is positive and that the node numbering is counterclockwise ensures that the Jacobian of the coordinate transformation is invertible at every point within the element. 1 . We consider a second order real elliptic Dirichlet problem in a bounded plane smooth domain Lu f in fi R2 y u 0 on dQ. In this chapter step by stepintegration procedures applied to dynamic analysis of Walentyski R. 409 474. 10 Calculation of element stresses 70 3. Together with the second and third self contained volumes 0750663219 and 0750663227 The Finite Element Method Set 0750664312 provides a formidable resource covering the theory and the application of FEM including the basis of the method its application to advanced solid and structural mechanics and to computational fluid dynamics. Discrete graphics can be seen in Figure 6. The AB bar is aluminum and its cross section area is is 200 mm2. 4 Stiffness matrix formulation 87 3. 2 Some Elementary Examples 223 6. Change of variables during integration. 5 of Chap. 4 Approximation of Domain and Boundary Conditions 192 5 STABILITY 205 5. 5 higher order quadrilateral elements 7. e for problems involving curved boundaries a family of elements isoparametric elements are used Write down the shape functions for 4 noded rectangular element using natural cordinate system. A Ciarlet Raviart type isoparametric mixed finite element method MFEM is constructed and analyzed for solving a class of fourth order elliptic equation with Navier boundary condition defined on a curved domain in R 2 and numerical quadrature is also considered in the scheme. 1 Introduction. It not only explains the theory but also presents the implementation of the theory into computer code. Non Isoparametric Isoparametric 2D Triangular Elements 4. Stack Exchange network consists of 176 Q amp A communities including Stack Overflow the largest most trusted online community for developers to learn share their knowledge and build their careers. Virtual Work Basis of Finite Element Method Triangular elements and numerical integration Quadrilateral elements and isoparametric formulations With isoparametric elements numerical fluxes along curved boundaries are computed much more accurately due to the high order representation of the computational domain. It is not pos sible for more complicated problems 2D 3D thus it is neces sary to utilize numerical integration. It has six unknown displacement degrees of freedom u1 v1 u2 v2 u3 v3 . 19 Numerical Examples of Isoparametric Elements 191 8. Panneerdhass lakshmi publications The Cholesky s method unlike the Doolittle s and Crout s does not have any condition for the main diagonal entries. Each element node has 3 degrees of freedom DOF DOF 1 and 2 for solid displacement u and DOF 3 for fluid pressure p . Superconvergence and Reduced Integration in the Finite Element Method By Milos Zlamal Abstract. The finite element formulations are developed through easy to follow derivations for the analysis of plane stress or strain and axisymmetric solid plate bending three dimensional solid and shell problems. The curved boundary is approximated by using isoparametric finite elements. In many cases Gauss formula is used K A Xm i 1 tD wi 22 where wi is weight of integrational point m is number of inte grational points. shear locking. Hexa20 Twenty node isoparametric solid brick element based on quadratic interpolation and numerical integration. 5. Introduction. 2 by using the two point Gauss quadrature rule is most nearly A 11. The integrand in this case is a matrix with constant terms independent of element coordinate system hence evaluation of this integral is trivial and in fact the integral can be evaluated exactly without the need of any numerical integration scheme. The TE12L element Fig. 2 General nite element solution 57 2. View L10 Isoparametric and NumInt1D. To facilitate the numerical integration the geometry of the element is expressed by interpolating functions using the natural coordinate system r s t of the element with its origin at the quot center. One method used to construct the lumped force stiffness matrices employs the trapezoidal numerical integration scheme which places the element nodes at integration points. These elements can be classified based upon the dimensionality ID II D and III D Elements or on the order of the element Lower order and Higher order elements . They identified that a dense mesh can result in an increase in precision however with a sacrifice of computing time. The Finite Element Method for Static Linear Elasticity. Computational examples and related MATLAB codes will be provided throughout to illustrate the algorithmic realization of the Finite Element Method. 18. 1 QUADRATIC ISOPARAMETRIC ELEMENTS 1 D A quadratic line element with three equally spaces nodes in natural coordinates 11 is considered Fig. . Which method of approach is useful for evaluating four noded quadratic elements a Numerical integration b Penality approach method c Gaussian quadrature approach d Rayleighs method View Answer This element is implemented for simulating dynamic response of solid fluid fully coupled material based on Biot s theory of porous medium. As such element have yet to be able to follow prescribed boundaries to a good degree of approximation curved shapes are desirable. 57 Write the Gauss points and weights for two point formula of numerical integration. Element integrals NEFEM also requires to compute integrals in an element e with one side e on the NURBS boundary see Figure 206 Mapped elements and numerical integration be greater than 1 800. by a zero level of a level set function accurate numerical integration is di cult. Finite Element Formulation. Although the use of triangular element can always fit a complicated boundary the accuracy of this element is low in general. Nagaraja currently serves as Professor at department of Mathematics Amrita School of Engineering Banglore campus. These Numerical Integration MCQ Questions Will help you to improve your Finite Element Method knowledge and will prepare you for various Examinations like Competitive Exams Placements Interviews and other Entrance Exmaniations Finite Element Method Multiple Choice Questions on Numerical Integration . 1002 nme. No. Evaluate the integral 22 10 2 3 x ydxdyby numerical integration. In this paper the same integrations are performed analytically Chapter 7 Two Dimensional Isoparametric Elements. 11 Numerical Integration 161. Often it is not feasible to solve the integrals analytically for an element in its most general appearance. 4 Three dimensional isoparametric elements 373 Numerical integration. Give the interpolation function associated with Node 5 midside between 1 and 2 in the Q8 element. In this paper the Isoparametric formulation for simple bar element is explained first and it will lead to a simple mathematical computation. In addition finite element specific aspects like the isoparametric concept or numerical integration will be addressed. Inthispaper weincorporatecell wisestrainsmoothing operations into conventional nite elements and propose the smoothed niteelementmethod SFEM for2Delasticprob 154 CHAPTER 6 Shape Functions Derivatives and Integration the number of nodes of the triangle with the same order. CPS4I and 2 2 2 points in three dimensions e. For instance a sti ness elemental matrix coe cient is computed as Ke ij Z Natural co ordinate systems Isoparametric elements Shape functions for iso parametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction to Analysis Software. The isoparametric formulation is generally applicable to 1 2 and 3 dimensional stress analysis. It is based on linear interpolation and numerical integration. Maryland Baltimore Md. Then we will illustrate the adaptability of the isoparametric formulation to common numerical integration methods. 2Quadratic Elements 291 5. FE terminology analysis steps and input output. Isoparametric finite elements 27 Many traditional isoparametric elements only have position coordinates Displacements are expressed in terms of the natural local coordinates and then differentiated with respect to global coordinates. Isoparametric Representation An isoparametric coordinate system has been used to evalu ate the shape functions of the parent element and map them to a generalised triangular element within the mesh. Computer Methods in Applied Mechanics and Engineering 102 2 255 289. via Universit 12 I 43121 Parma. A FEM for unsaturated transient seepage is established by using a quadrilateral isoparametric element considering the fact that the main permeability does not coincide with the axis situation. Numerical Integration MCQs Here you will find MCQ Questions related to quot Numerical Integration quot in Finite Element Method. 5 A finite element procedure for the large deformation dynamic response of axisymmetric solids The element level load vector due to transverse load q per unit area is obtained 9. The integration is the reduced second order to prevent shear locking. Penta6 Six node isoparametric solid wedge element based on linear interpolation and numerical integration. Download n_i n_n n_g number of DOF per element MODEL input file controls for various data generators pre_e 0 Dummy items before el_type amp connectivity Isoparametric quadrilateral elements Shape functions for rectangular elements Isoparametric mapping for quadrilateral elements Numerical integration for quadrilateral elements Four node quadrilateral element for 2D BVP Eight node serendipity element for 2D BVP Isoparametric triangular elements Natural or Area coordinates for triangles Integration using gauss integration formula in 8 the default or 27 integration points. Huerta NURBS Enhanced Finite Element Method NEFEM International Journal for Numerical Methods in Engineering 76 1 56 83 2008 Differentiation and Integration 24 Square Matrix 25 7 TWO DIMENSIONAL ISOPARAMETRIC ELEMENTS AND NUMERICAL INTEGRAnON 7. With the help of the Galerkin weighted The coefficient matrices of isoparametrically distorted finite elements are frequently computed by numerical integration formulas with a lower degree of exactness than required by the degree of the integrand. variables using isoparametric elements for the spatial discretization 2nd to 4th order and time integration is explicit and based as in DP on 2nd order truncated Taytor series expansions. The methods are a necessity for quadratic elements as well as higher order elements. e. The finite element method is exactly this type of method a numerical method for the solution of PDEs. Element Stiffness Matrix 276. Understand some applicable considerations and advanced applications of FEM. Isoparametric elements. NG 1 2 3 Polynomial order 2 NG 1 1 3 5 Curve y x Linear Cubic 5th order 269 NUMERICAL INTEGRATION REVIEW For double integration If 2 Gauss pts. The book includes chapters on solving finite element equations and numerical integration and a chapter on programming finite element method 1993 scC1 curved finite elements with numerical integration for thin plate and thin shell problems part 1 construction and interpolation properties of curved scC1 finite elements. 7 three dimensional isoparametric elements 7. The second concept is that of numerical integration. 56 When do we resort to numerical integration in 2D elements AU Nov Dec 2013 5. Their local coordinates are xed and determined such that the numerical integration is as Finite Element Modelling Techniques in MSC. Theintegration order required to evaluate a specific element matrix exactly can be evaluated by study ing thefunction f to be integrated. 7. 1 Isoparametric Formulation Element stiffness matrix http ocw. 8 finite element modeling practice for isoparametric elements 7. 2 The Four Node Quadrilateral 273. element equations matrix form to solve numerically . In the next Chapter these techniques combined with numerical integration are applied to quadrilateral elements. If the reverse is true the element is referred to as superparametric. P. 1 Introduction 208 7. are issues of numerical integration and sampling. 2 The Condition Number 209 6 EIGENVALUE PROBLEMS 216 6. Master isoparametric element and numerical integration methods of FEM. 1 and x 1. 4. A Textbook of Finite Element Analysis presents detailed finite element formulation for 1D 2D 3D trusses beams frames plane stress plane strain axisymmetric 3D problems plates and shell problems. This set of Finite Element Method Multiple Choice Questions amp Answers MCQs focuses on Numerical Integration . Element has parametric stiffness matrix tranform for depend to Jacobian transform partial isoparametric formulation. hourglassing D. isoparametric plate element. I realized isoparametric CST 3 node triangle with 1 Gauss integration point and LST 6 node triangle with 3 Gauss int. His areas of research include Numerical Analysis Finite Element Method Transforms and partial differential equations. Abdirhamen Shibru. Through thickness displacement and stress fields at the finite element discrete nodes are observed to be in excellent agreement with the elasticity solution. 1 1 ii Derive the element strain displacement matrix of a triangle element. Finite Element Analysis FEA is a practical application of the Finite Element Method FEM for predicting the response behavior of structures or fluids to applied We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed. Previous papers in this area 6 7 dealt chiefly with element stability through eigenvalues and the stiffness matrix trace. 1 6. It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. 5. Motivation The linear triangle presented in Chapter 115is an isoparametric element although was not originally Isoparametric Numerical Integration on Triangular Finite Element Meshes January 2004 Comptes rendus de l 39 Acad mie bulgare des sciences sciences math matiques et naturelles Finite Element Method and the Analysis of Systems with Uncertain Properties Formulation and calculation of isoparametric finite element matrixes 1. Beams and Frames. present numerical examples which demonstrate the quality and potential of this method. 1 Gauss Quadrature points and weights for 2D quadrilateral elements NGP 1 0 0 2 4 NUMERICAL ANALYSIS OF ENGINEERING STRUCTURES LINEAR ELASTICITY AND THE FINITE ELEMENT METHOD Author Tam s Mankovits Reviewed by Zsolt Tiba 2014 Numerical approximation of elliptic interface problems via isoparametric finite element methods. 13. The wide controversy due to conventional element level approximate numerical quadrature within the computational square domain in and coordinates is completely resolved here by the closed form analytical integration within the home reference library technical articles manufacturing and process equipment 14. Express the role of numerical integration in the finite element method. The reaction diffusion equation on curved domains is considered. UNIT VIII. In other Additional notes integration by parts intgeral formula sheet Additional notes more information about weak forms 2 Time for programming 1D 3 Numerical integration and isoparametric elements Presentation quot Numerical Integration quot B. 3 Family of isoparametric elements 83 3. 8 Choices in numerical integration 1 Full integgq gration quadrature that is exact for rectangular geometries How many do you need for Q4 element Reduced integration may remove resistance to some displacement modes For example one point integration of Q4 element removesFor example one point integration of Q4 element Numerical Methods 20 Multiple Choice Questions and Answers The hypothetical machine has two I O instructions 0011 Load AC fro I O 0111 Store AC to I O Solutions List and briefly define two approaches to dealing with multiple interrupts List and briefly define the possible states that define an instruction execution applications an isoparametric representation of the geometry is far from providing an optimal numerical solution for a given FE discretization. Table 3. 8 TWO DIMENSIONAL ISOPARAMETRIC ELEMENTS. Use 3x3 Gauss integration For rectangular elements J const Consider the following element Note here we could use h i x y directly. Part II Strong and weak form of heat transfer in 2D and 3D. For a four noded quadrilateral elements In u T u. 3 Numerical Integration 279. Only parametric integral transformed isoparametric integration. As seen in Section 6. 1 The Calculation of Shape Functions 4. A. Solve an Engineering problem using finite element software and to test the accuracy of the result by comparison with an analytical solution. Isoparametric map. 2008 . Thus the shape functions for a six node triangle may be obtained using quadratic order polynomials as All elements use numerical integration to allow complete generality in material behavior. Thus numerical methods must be used. it The quadrilateral isoparametric element is used to discretize the dam body of which the body contains a total of 312 units and 351 nodes. Isoparametric Representation 16. K. Hence the evaluated stiffness matrix is exact for the choice of finite element here. Finite elements III 2D boundary value problems. In the recent paper 35 a new approach based on isoparametric mappings of the underlying mesh outlined 3. In commercial software the isoparametric formulation is used for both low order To clarify the proposed model of the subsoil a method is derived for determination of vertical subsoil stress analysis under any shape of a slab construction by means of numerical integration and theory of isoparametric elements using the Jacobian transformation. 2 can be used for GQ integration of triangular elements. XE quot Numerical Integration Rules 14 Point 3D Rule quot First consider a three dimensional 14 point numerical integration rule that is written in the following form I Numerical integration Iso parametric Mapping The actual quadrilateral element in Figure 1 a is mapped into the parent element in Figure 1 b . To be able to apply multigrid methods a sequence of fi Stress strain analysis of elastic halfspace by means of Gauss numerical integration and Jacobean of transformation is presented. Transformation from natural coordinate system to real coordinate a and numerical integration in natural coordinate system of a 4 node element region b . Numerical integration with Galerkin method Numerical integration is a central process with the standard Galerkin method 10 . 3 7. 2 Preprint of R. 9. 2 Gauss Quadrature Plate and general shell element Target where by using numerical integration instead of explicit integration where t ij thickness at the integration point ij weighting factor Integration points r i s j to evaluate F ij of a 4 node plane stress element B ij J ij C are unknowns 1 r 1 1 s 1 Volume evaluated in second order self adjoint eigenvalue approximations using classical isoparametric nite element methods has been estimated 5 and 6 being the papers which deal with the e ect of only numerical integration on eigenvalue approximations. C. 1 Variational Formulation and the Min max Principle 216 6. 3 Four Node Rectangular Element 153. 16. square for any quadrilateral element Advantages include more flexible shapes and compatibility NUMERICAL INTEGRATION. Week 11 Plane strain and plane stress problems. necessary to employ numerical integration to evaluate the element stiffness matrix. We prove that iffhas third order The isoparametric shape functions for CST element are N1 N2 and N3 1 . Which method of approach is useful for evaluating four noded quadratic elements a Numerical integration b Penality approach method c Gaussian quadrature approach d Rayleighs method. finite element space denned over Dh a union of elements has approximation power h k in the L 2 norm and if the region of integration is approximated by Q with dist Q Ch k then it is shown that one retains optimal rates of The element integrals as described in the previous chapter can be solved analytically or numerically also called direct respectively numerical integration. 10 exercises chapter 8 finite element analysis Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations ODEs . l Numerical integration We will calculate the stiffness matrix of this approach as the homogeneous numerical integration HNI method. The existence and uniqueness of the numerical solutions are proved The isoparametric shape functions for CST element are N1 N2 and N3 1 . Especially in a setting where geometries are only implicitly described e. email protocollo pec. Subject Numerical Methods and Optimization Semester II Tip for mobile user If file is opened in preview mode use download option given in right side top corner three dot menu to download the PDF file MIT 39 s Department of Mechanical Engineering MechE offers a world class education that combines thorough analysis with hands on discovery. Numerical integration in the computational element h e approximation of e is performed using the isoparametric transformation given in Equation 2 with a numerical quadrature in the reference element I. For its numerical solution we employ the finite element method with quadratic triangular Finite element analysis Isoparametric element. The basic idea is to interpolate the element geometry from the coordinates of the nodes. Isoparametric Elements MCQs Free download as Text File . elements or with the use of elements with more complicated shape functions. Chapter 4 Unit V Isoparametric Formulation Natural co ordinate systems Isoparametric elements Shape functions for isoparametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction 5 One Dimensional Elements 287 5. Numerical integration in finite elements Integration of expressions for stiffness matrices and load vectors can not be performed analytically for general case of isoparametric elements. 1 Violations of the Rayleigh Ritz Code 4. 11 11 kBDBhJdsdtT Isoparametric Elements Evaluation of the Stiffness Matrix by Gaussian Quadrature A flowchart to evaluate k for an element using four point Gaussian quadrature is shown here. 18 Vector Computation in Isoparametric Elements 187 8. Time Dependent Problems 6. Finite element modelling of time dependent problems dynamicproblems Isoparametric representation finite element modeling of axisymmetric solids modeling of orthotropic material system Four node quadrilateral elements Numerical integration Higher order elements conjugate gradient implementation of the quadrilateral elements Natural coordinates shape functions of constant and linear strain triangular elements isoparametric mapping examples L13 Triangular elements 2 Evaluation of element matrices the Jacobian matrix examples of specific cases. The purpose of the isoparametric formulation is to create shape functions that would ensure the compatibility of the displacement between neighboring elements while maintaining the requirements for shape functions mentioned in the previous section. In the isoparametric element coordinate transformation and displacement function always adopt the same nodes. 2 The resulting DG weak formulation to be discretized at each element e is Z e W dUe dt Z e W xk 2. Programming of the procedure of calculation of stiffness matrix. The extended finite element method XFEM is a numerical technique based on the generalized finite element method GFEM and the partition of unity method PUM . b Explain the application of Gaussian numerical integration with n 2 each way. Weaknesses of procedural approach examples of procedural oriented FEM codes. For solution the modified Newton iteration in equation 26 is probably in most analyses most effective. txt PDF File . Numerical integration in ngsxfem Numerical integration on cut elements with lin mesh transformation h Integration on linear level set domains on simplices Geometrical decomposition into simplices Integration by transformation to reference domain lin Transformation factors det D h are automatically considered due to Numerical and Computer Methods in Structural Mechanics is a compendium of papers that deals with the numerical methods in structural mechanics computer techniques and computer capabilities. The basic element useful for two dimensional analysis is the triangular element. For example the closed form result for the simple truss element is Evaluation of the stiffness matrix integral for the isoparametric formulation requires the use of a numerical integration process such as the Gauss Quadrature method to evaluate the terms in the stiffness integral. F C0. 13. NASTRAN and LS DYNA 6 1 FINITE ELEMENT MODELLING TECHNIQUES AND MSC. 2 Non conforming Elements 174 4. Load Vector. 2 The Strain Displacement Matrix 4. A large number of finite elements may be used to obtain reasonable resemblance between original body and the assemblage. Triangular and quadrilateral shape functions. 3 BASICS OF FEM 65 3. In two dimensional isoparametric elements we can generate element stiffness matrix by using ____ a Numerical integration b Differential equations c Partial derivatives d Undefined The isoparametric shape functions for CST element are N1 N2 and N3 1 . 2 3 Preconditioning isoparametric finite element methods taking into account numerical integration Furthermore numerical integration 16 for solving one and two dimensional integrals is explained in connection with the application of the finite element method. 10. Isoparametric elements will be considered in more depth later. edu. Loads Body loads are supported. Analysis of Beams and Rigid Frames 242 14. L14 Triangular elements 3 Numerical integration examples of numerical evaluation of element matrices L15 Beam elements 1 which four node bilinear quadrilateral isoparametric elements are employed with 2 x 2 Gaussian quadrature rule for the numerical integration of stiffness matrices and force vectors. More details about this numerical model can also be found in Grilli 1993 . 54. Zienkiewicz O. Concept of isoparametric elements Terms isoparametric super parametric and subparametric. Fern ndez M ndez and A. 1. 7 Numerical Integration 230 13. STRUCTURAL ANALYSIS WITH THE FINITE ELEMENT METHOD Linear Statics Volume 1 The Basis and Solids Eugenio O ate The two volumes of this book cover most of the theoretical and computational aspects of the linear static analysis of structures with the Finite Element Method FEM . 3. a b Figure 3. The theory of the Finite Element Methods and several widely used analytical techniques. 7 isoparametric elements and numerical integration Biomechanics Concepts and Computation Using a quantitative biomechanics approach this book integrates both general and specific topics theoretical background and biomedical engineering applications as well as Natural co ordinate systems Isoparametric elements Shape functions for iso parametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction to Analysis Software. 1 . Basics of Numerical Di erentiation Richardson Extrapolation Basics of Numerical Integration Quadrature Formulas By writing the strain field as a non local weighted average of the compatible strain field integration on the surface of the finite elements is transformed into boundary integration so that the usual subdivision into integration cells is not required an isoparametric mapping is not needed and the derivatives of the shape enrichment MULTIPLE CHOICE QUESTIONS Integration. Finite Element Method Various element formulations for metallic and composite structures isoparametric element formulation Numerical Integration concept of consistency In the Finite Element Method we use several types of elements. We recall the approximation for the solution of the wanted variable from Eq. A . The number of integration points within a layer at any given section point depends on the element type. full integration we will discuss the method of obtaining the stiffness matrix for a practically important class of finite elements in a plane state stress strain in isoparametric approach so called isoparametric elements. 1 3. l We will derive the stiffness matrix of simple beam elements and rectangular elements using an iso parametric formulation. 1Linear Elements 287 5. In the previous chapter the shape functions N i have hardly been discussed in any detail. Introduction In finite element applications the derivative of the approximate solution is frequently a quantity of primary interest. The four interpolation functions associated with the nodes are N1 1 4 1 1 2 1 N 2 1 4 1 1 2 1 N3 1 4 1 1 2 1 N 4 1 4 1 1 2 1 The element has 4 integration points which are shown in the gure below. 2 of Silvester amp Ferrari 2nd Edition Natural co ordinate systems Isoparametric elements Shape functions for iso parametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction to Analysis Software. 16 Patch Test 179 8. Construct execute and interpret linear structural finite element models. 5 Fifteen node isoparametric solid wedge element based on quadratic interpolation and numerical integration. NASA Technical Reports Server NTRS 19760009723 BOPACE 3 D addendum The Boeing plastic analysis capabilities for 3 dimensional solids using isoparametric finite elements Item Preview remove circle Logah Perumal and Thet Thet Mon Generalized equations for numerical integration over two dimensional domains using Quadrature rules Integration Mathematical Theory and Applications Vol. V. Hence the integral is approximated by the weighted sum of Body loads are supported. Numerical integration method is selected Gauss Legendre Numerical integration Method. Numerical integration using Gauss quadrature. 8 Examples 35 isoparametric four node finite element showing accurate results for a distorted element. We will formulate and integrate numerically in time the 1D elastodynamics equations for transient dynamic analysis and discuss modal analysis. 57 The stiffness matrix K e and M e are evaluated first by expressing the integrals in the local isoparametric coordinates and of the element and then performing numerical integration employing the 2x2 Gauss quadrature. structral member areas. Master general principles and formats for elastic problems of FEM. element in Cartesian coordinates see Equation 2 . The element matrices are assembled Relate element order to shape functions isoparametric mapping to mesh quality. To improve computational efficiency it has become a practice of some analysts to 39 underintegrate 39 the various element stiffnesses. Each i th element side begins in i th element node and ends on next element node i 1 th node or 1 st node in the case of side number 3 . 3 Numerical Integration 4. Similar to the thermal energy conservation referenced above it is possible to derive the equations for the conservation of momentum and mass that form the basis for fluid dynamics. Abstract An increase of available parameters associated with an element usually leads to improved accuracy of solution for a given number of parameters representing the whole assembly. at each integration point. Discrete weak form with linear triangular elements and resulting system of equations. Week 12 Coordinate transformations into integration elements. This BVP is then transformed into a set of initial value problems IVPs so as to use the available efficient and effective numerical integrators for them. Their use is also known as quot numerical integration quot although this term can also refer to the computation of integrals. One prerequisite for numerical integraton is an expression for rate of change of length area or volume with respect to intrinsic coordintes An improvement of the classical finite element method is proposed. php Isoparametric elements enable meshing an irregular domain with triangular or quadratic elements that do not maintain orthogonality between the sides of the element. For elements not intersecting the boundary a standard finite element interpolation and numerical integration is used. The analysis of thin walled structures requires a reduction from three to two dimensions. Apply numerical integration. Natural co ordinate systems Isoparametric elements Shape functions for iso parametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction to Analysis Software. Lachat and Watson Ref 5 present a variable 8 TWO DIMENSIONAL ISOPARAMETRIC ELEMENTS. The normal mode method is well known in the litera ture and is described briefly in Section 4. Among finite elements the curved isoparametric elements of the Serendipity family see Zienkiewicz 8 are mostly used in the finite element codes prepared for engineering computations. 6 volume integrations as well as surface integrations should be carried out in order to represent the elemental o Numerical integration in finite element analysis At each integration point of the element the shell wall is divided into a number of stations with constant intervals T through the thickness. 5Isoparametric Elements 298 5. The polynomials for the translations u xyz can be expressed as Victor E. Transformation to an s t coordinate system a natural coordinate system facilitates integration. nctu. By writing the strain field as a non local weighted average of the compatible strain field integration on the surface of the finite elements is transformed into boundary integration so that the usual subdivision into integration cells is not required an isoparametric mapping is not needed and the derivatives of the shape enrichment Derive the terms for isoparametric finite elements and transform information between the two coordinate systems. Stefan Schmalholz ETH Zurich Numerical modeling of rock deformation 11 FEM 2D numerical integration amp isoparametric elements Stefan Schmalholz schmalholz erdw. numerical integration in the same manner as f s t was integrated. Finally we will consider some higher order elements and their Another important step in finite element analysis is the integration over the elements. Numer Integration 8 functions there is tendency to lock too stiff numerical behavior Reduced integration is the most basic engineering approach to resolve this problem Mathematically more rigorous approaches Mixed variational principles based e. 13 Stress Refinement and Stress Smoothing 168 8. Isoparametric Representation 16. 4 No. 6Numerical Integration Gauss Legendre Quadrature 300 5. bending deformation C. Introduction presented. These Linear Algebra Multiple Choice Questions MCQ should be practiced to improve the Data Science skills required for various interviews campus interview walk in interview company interview placements entrance exams and other competitive examinations. MANE 4240 amp CIVL 4240 Introduction to Finite Elements Isoparametric elements in 1D and Numerical displacement shape functions the element is called subparametric. This is the reason why we shall rst become familiar with nite element approximation before addressing nite element methods for di erential equations. f for each integration point p find the Jacobian dv at undetermined point xi p find weight corresponding to undetermined point xi p find det jacobian weight corresponding to point xi p find B matrix at the undetermined point xi p for i 1 number of ponts in element for j number of points in element K i j dv B i B j The sequence of nodes defining the variable node isoparametric element is shown in Fig. Guass elimination Numerical integration. Finally we will consider some higher order elements and their associated shape functions. Numerical integration techniques can be used with ease. ch. Equation System of equations FEM equation definition. That is how to obtain a numerical value of where can be a 1D 2D or 3D domain. A brief discussion is included on the condition of the stiffness matrix obtained with such elements and the accuracy of the numerical integration. 17 isoparametric FE is required. The addition of incompatible displacement modes to isoparametric elements in 1971 was an important but minor extension to the formulation 5 . You can define the element stacking direction to be along one of the element 39 s isoparametric directions see Figure 5 for element stack directions . 1 Patch Tests as Numerical Experiments 6. It has to be emphasized that the coordinate transformation is done for calculation purposes of numerical integration only. Then the quadrature rules for the numerical integration of the stiffness matrix and the equivalent nodal force vector for triangular and quadrilateral elements are explained. 3 Analysis of Rigid Plane Frame Using 2 Noded Beam Elements 259 14. sional elements. 32 0 obj De ne basis and The second order wedge uses 9 integration points for its stiffness matrix but 18 integration points for its consistent mass matrix. Derivation and implementation of a basic 2D FE code with triangular constant strain elements Generalization of finite element procedures for linear elasticity interpolation and numerical integration in 1D 2D For updates and downloads Visit the VisualFEA Center. By writing the strain field as a non local weighted average of the compatible strain field integration on the surface of the finite elements is transformed into boundary integration so that the usual subdivision into integration cells is not required an isoparametric mapping is not needed and the derivatives of the shape enrichment To submit an update or takedown request for this paper please submit an Update Correction Removal Request. unipr. ME6603 Finite Element Analysis III Year VI Semester SRI ESHWAR COLLEGE OF ENGINEERING Coimbatore 202 Department of Mechanical Engineering WorkSheet Quiz S. tel. We consider a second order real elliptic Dirichlet problem in a bounded plane smooth domain Lu f in fi Ry u 0 on dQ. Raviart The combined effect of curved boundaries and numerical integration in isoparametric finite element methods The mathematical foundations of the finite element method with applications to partial differential equations Proc. Derive the stiffness term k46 for the isoparametric Q4 element until the application of numerical integration. 3 Numerical Integration 181 4. 2 Element stiffhess matrix and element force vectors 23 2. In practice the integration is frequently not performed exactly butthe integration order must be high enough. Parametric method for plate element analysis than isoparametric applications is more complex. 2. Usually equation 6 leads to some collocation points located on the boundary of each . The formulations of several element types isoparametric formulation of the elements stiffness matrices and numerical integration. 17 Triangular Tetrahedral and Prismatic Curved Side Elements 183. 2 The Condition Number 6. The isoparametric shape functions for CST element are N1 N2 and N3 1 . 3 two dimensional isoparametric quadrilateral element 7. 1 Ju with U th unie t spher ine R3. In the case of straight sided isoparametric triangular elements numerical integration is exact provided a suf cient number of integration points are used. MR 0421108 CE 546 ME 549 Finite Element Analysis Syllabus for CE 546 ME 549 Text A First Course in Finite Elements by Jacob Fish and Ted Belytschko published by J. For continuum solid shell elements the material layers can be stacked only in the third isoparametric coordinates parallel to opposite faces 1 and 2 of the isoparametric master element. The design realization and analysis of un tted nite elements methods which are higher order accurate is a challenging task. 2014 Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher Order Curved Triangular Elements. 7 Isoparametric Triangular Elements 178. Notice this analysis is NOT ISOPARAMETRIC MODEL. We will look at the development of development of finite element scheme based on triangular elements in this chapter. Answer c Numerical modeling of rock deformation FEM 2D Elasticity. 5 8 node isoparametric quadrilateral element Q8 30 2. Also the possibilities you have to inspect and modify the numerical integration schemes in the COMSOL Multiphysics software are highlighted. For two dimen In the present Chapter the concept of isoparametric representation is introduced for two dimen sional elements. 4 Isoparametric Finite Elements and the Four Node Quadrilateral 4Q Element 158. 4 Higher Order Defining the stacking and thickness direction based on the element isoparametric direction. The key issue of XFEM including constructing displacement pattern establishment of the governing equation and scheme of numerical integration is also presented. We will derive shape functions and discuss numerical integration. 6 is safer 6. i e. 1 Introduction 80 3. of Gauss pts. 17 Triangular Tetrahedral and Prismatic Curved Side Elements 183 8. Using a standard Galerkin method the governing equations of a given problem are transformed into the following classical variational form Home Browse by Title Periodicals Applied Mathematics and Computation Vol. This is mainly due to the fact that e cient highly accurate numerical integration on domains that are implicitly described is not straight forward. x L An Introduction to Elasticity and Heat Transfer Applications The integration is the reduced second order to prevent shear locking. NUMERICAL INTEGRATION Numerical Integration is of prime importance when we deal with Finite Element Analysis especially in case of ISOPARAMETRIC ELEMENTS. Finite elements with linear shape functions produce exact nodal values if the sought solution is quadratic. The integrals expressed by isoparametric elements usually cannot be calculated through analytical methods because they are rather complicated. For a triangular element master element coordinates do not vary between 1 and 1 and we need to use a completely different GQ table specifically designed for triangular elements. Transient Behavior Mass Matrices 2 6 Lecture 2015 2014 g. Boundary loads are supported and are computed using numerical integration. But the situation is still worse in the case of isoparametric mixed nite element analysis of In order to overcome this drawback isoparametric elements are used i. THE FINITE ELEMENT METHOD PART II 7. The key purpose of this chapter is first to introduce isoparametric shape functions and second to outline numerical integration of the integrals appearing in the element coefficient matrices and element column. Some papers discus the analytical basis of the computer technique most widely used in software that is the finite element method. 3 CHAPTER 6 ISOPARAMETRIC ELEMENTS amp CHAPTER 7 ISOPARAMETRIC TRAINGLES. ethz. 4Global Local and Natural Coordinates 296 5. 1 Isoparametric Formulation Element stiffness matrix 10. Superconvergence of the tangential derivatives at Gauss points on edges of elements is proved for these nonlinear problems and quasiuniform partitions of the domain. Wilson represents a specific individual material embodiment of a distinct intellectual or artistic creation found in Boston University Libraries. 1972 Academic Press New York 1972 pp. 3 Equivalent domain integral EDI method 59 2. Using D the matrix M is evaluated at the start of each increment and for each of the stations. Numerical integration using Gauss quadra ture. Di erent techniques have been siders e cient strategies for numerical integration on elements a ected by curved boundaries. Further explicit integration cannot be carried out for the six node triangle and the eight node quadrilateral. isoparametric element Elements that use the same shape functions interpolations to define the geometry as were used to define the displacements. The default number of integration point used can be overloaded using NIP parameter. The isoparametric family includes elements for plane solid plate and shell problems. However the current paper is a more comprehensive study of these elements under prac Isoparametric elements numerical integration algorithms of calculation of stiffness matrix. 0 Two Dimensional FEA Frequently engineers need to compute the stresses and deformation in relatively thin plates or sheets of material and finite element analysis is ideal for this type of computations. Derive the stiffness term K56 for the isoparametric Q4 element in the figure using Gaussian numerical integration with n 2 each way. Although there are different numerical schemes available Gauss Quadrature is most ideally suited for isoparametric to define the shape functions and 2 the integration over the surface of the element is easier we will use numerical integration which is much simpler in the natural coordinate systems and can be scaled to the actual area The steps in deriving the elemental stiffness matrices are the same Step 1 Select element type Step 2 Select displacement The first one is that of isoparametric interpolation. The cartesian cordinates of interior point Point P are 3 6 . Element Properties. Quad4 Four node quadrilateral isoparametric plane stress element based on linear interpolation and numerical integration. What is LST element Isoparametric Elements Introduction Next we will introduce numerical integration methods for evaluating the quadrilateral element stiffness matrix. This integration follows the normal values for Gauss quadrature and a standard table Appendix F of Chari amp Salon table 7. 7. 6 In transformations based on parabolic type serendipity func tions it is necessary in addition to this requirement to ensure that the mid side nodes are in the middle half of the distance between adjacent corners but a middle third shown in Fig. Coordinate mapping Natural coordinates Area coordinates for triangular elements higher order triangular and quadrilateral elements Lagrangean and serendipity elements geometry associative mesh quality checks mesh refinement p vs h refinements Incompatible mode elements use full integration. 2 Four node isoparametric quadrilateral element 366 19. for problems involving curved boundaries a family of elements known as isoparametic elements is used 78 Write down the shape function for 4 noded rectangular elements using natural co ordinate system. Taking the derivative of this expression to find the Jacobian terms integration from the direct application of the Gauss rules. 672 B 11. Dr. 1 is needed such as for the 20 node element of the unmodified Degenerated Isoparametric Shell Element. Obviously there is danger in using commercially tetrahedral finite elements. 4 Triangular elements area coordinates numerical integration. It is an extremely The fundamentals of the Finite Element Method and modeling techniques as applied to structural analysis problems. QUADRATIC ISOPARAMETRIC ELEMENT FOR PLANE ELASTICITY 80 3. Eigenvalue Problems COMMENTS 1. Walentyski R. Solution Techniques Numerical Integration Static condensation assembly of elements and solution techniques for static loads. Hexa8 Eight node isoparametric solid brick element based on linear interpolation and numerical Keywords finite element analysis body force weight force numerical integration Gaussian quadrature Highlights Efficient method for the calculation of body forces within finite element analysis was implemented. . To avoid this effect the element matrices may be stabilised i. The 8 node hexahedron continuum shell has three possible stacking directions the 6 node in plane the element sti ness matrix are assembled into the global column and matrix. In order to overcome this drawback iso parametric elements are used i. com. Download with Google Download with Facebook. What is CST element Three noded triangular elements are known as CST. Numerical Integration is of prime importance when we deal with Finite Element Analysis especially in case of ISOPARAMETRIC ELEMENTS. Introduction to Variational Methods Weak form of Governing Equation Weighted residual method Introduction to Finite elements and Galerkin Finite elements. article Bhattacharyya2002 abstract Estimates for the combined effect of boundary approximation and numerical integration on the approximation of simple eigenvalues and eigenvectors of 4th order eigenvalue problems with variable constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method which in the particular case of bending problems 4. Isoparametric Serendipity Hermitian Elements 129 Basics of the Finite Element Method. Higher Order Elements. 390521902111. Convergence. This may result in a rank deficiency of the element matrices which causes an oscillatory instability in the solution. One of the original six courses offered when MIT was founded in 1865 MechE 39 s faculty and students conduct research that pushes boundaries and provides creative solutions for the world 39 s problems. Download finite element analysis by dr. 1 Restrictions on the mapping 364 19. on the Hellinger Reissner functional Reduced Integration Beam Elements Shape function Cubic order Quadrature condition for a four node isoparametric element is that no interior angle is greater than 180 and the positivity of Jaco bian determinant should be ensured in numerical implemen tation. The new set of sampling points is a mimic of Gauss numerical integration method. l It allows to create elements with a shape of a straight line or a curved surface. Kaus 2011 Presentation quot Isoparametric Elements quot B. The In this study the Kernel integration scheme for 2D linear elastic direct boundary element method has been discussed on the basis of subparametric element. Two schemes are offered full integration and reduced integration. Figure 6. B. It is worth noting that at nodes the nite element method provides exact values of u just for this particular problem . Shell and beam element properties can be defined as general section behaviors or each cross section of the element can be integrated numerically so that nonlinear response can be tracked accurately when needed. Gauss Quadrature formula is the most commonly used Numerical Integration schemes. or. Isoparametric quadrilateral elements Shape functions for rectangular elements Isoparametric mapping for quadrilateral elements Numerical integration for quadrilateral elements Four node quadrilateral element for 2D BVP Eight node serendipity element for 2D BVP Isoparametric triangular elements Natural or Area coordinates for triangles JNTU Hyderabad B. such as the isoparametric 5 node quadrilateral element and 3 node bar element. 4 Interaction integral method 59 Chapter 3 Extended Finite Element Method for Isotropic Problems 3. A review of these operations here is intended to serve as a background to a discussion of the different options that are availble for mrmerical analysis of shell structures. Numerical integration over the element volume area implies that a loop over integra tion points is entered. 7 Stress matrix 1O1 Shear locking defect remains addressed by the isoparametric Q6 element Other 2D isoparametric elements I ST Q8 amp Q9 Possibly to have initially curved sides due to the midside nodes These geometric distortions are usually detrimental to accuracy Very useful for modeling arbitrary geometry holes or fillets Numerical integration 3 Before we go into deeper detail of coordinate transformation a few terms have to be explained. Stability 5. Continuum Elements Part I Isoparametric Elements numerical Integration Chapter 10 HW 5 HW 4 6 5 Jun Midterm Exam 90 min Project Discussion HW 5 7 7 Jun Static Condensation Performance of Elasticity Elements Stress Calculations Barlow s points Continuum Elements Part II Section 4. Gauss integration Choose only number of integration points n position of integration points and weights not choosen Consider the polynomial of order 2n 1 Exact integration gives Numerical integration scheme is With g x as above we get 17. 2D Triangular Constant Gradient Elements 2 3 Lecture 2014 d. It creates a function by using the element amp x2019 s node hydraulic head and shape function instead of the real head in the Richard seepage control equation. Master various elements and shape function of FEM. This scheme locates the integration points at the element end An introduction to the finite element method with emphasis on fundamental concepts. 0 Linear Algebra MCQs Questions And Answers. ch LEB D3 HS2008 Thursday 10 12 CAB H 53 Numerical modeling of rock deformation FEM 2D Elasticity. 13 Consistent nodal loads stress calculations effects Find many great new amp used options and get the best deals for The Boundary Element Method with Programming For Engineers and Scientists by Ian Smith Gernot Beer and Christian Duenser 2010 Trade Paperback at the best online prices at eBay Free shipping for many products The isoparametric shape functions for CST element are N1 N2 and N3 1 . Isoparametric element and numerical integration. The integration formulae will include product Gaussian quadratur an somde e methods o basen breakind Ug into smaller triangular elements with various associated low order integration schemes. View Answer Chapter 6 Isoparametric elementsChapter 6 Isoparametric elements Same shapppe functions are used to interpolate nodal coordinates and displacements Shape functions are defined for an idealized mapped elt f diltll t lement e. Then the Numerical Integration for Gauss Quadrature for two and three sampling points is explained. If a nodal sequence different from Fig. 2 Isoparametric concept 81 3. IVA 00308780345. 4 7. i is called a Gauss point. Numerical illustrations for a convex quadrilateral and a triangle with a side node are included. 2 Non conforming Elements 4. Because we have the limits 1 and 1. AND NUMERICAL INTEGRATION 273. 8 Continuity and Completeness of Isoparametric element 39 locks 39 when it is thin assuming 39 full 39 numerical integration . FE Modelling degrees of freedom stress strain and strain displacement relations. Isoparametric elements cont d Mapping of elements 4 node bilinear quadrilateral Jacobian matrix Understanding of J and J Examples Stiffness matrix Q8 T3 T6 2 This Class Numerical integration Gauss quadrature Introduction to numerical integration and Gauss quadrature Gauss quadrature in one dimension 1D Finite Element Analysis structural engineering Finite Element Method FEM is a numerical technique for solving differential equations that describe many engineering problems. 2 Beam Analysis Using two Noded Elements 242 14. Preface This is a set of lecture notes on nite elements for the solution of partial differential equations. Noting that 20 in which J is the Jacobian matrix which can easily be evaluated by a numerical process 3 noting that Jl Isoparametric elements and numerical integration Computational strategies 4. 1994 . The most commonly used techniques for numerical integration are the midpoint rule trapezoidal rule and Simpson s rule. Visit the VisualFEA Center. 8. XFEM CutFEM TraceFEM Finite Cell fictitious domain more computational effort than numerical integration. 1 Isoparametric nite elements a short review 65 analysis of high order accurate un tted nite element methods is still challenging. Kaus 2011 for the Isoparametric quadrilateral element shown in fig determinethe local co ordinates of the point P whose Cartesian co ordinatesas 6 4 Explain the concept of numerical integration and its utility in generating Isoperimetric finite element matrices. Elements Formulation Numerical Integration Gauss quadrature one two and threedimensional triangular elements formulation rectangular elements Serendipity elements Illustrative Examples. 3 5 for this. Isoparametric formulation Concepts of isoparametric elements for 2D analysis formulation of CST element 4 noded and 8 noded iso parametric quadrilateral elements. Hence although the basic continuum Week 8 Shape functions for 1 dim. 13 Stress Refinement and Stress Smoothing 168. Week 9 Triangular elements Shape functions Numerical Integration. The isoparametric nite element Qr K b K Qr K d in quadrilaterals and hexahedra The rst di culty is to adopt a good mathematical de nition of a quadrilateral in 2D or an hexahedron in 3D. Shape Functions 273. Numerical modeling of rock deformation FEM 2D Elasticity. Numerical Di erentiation and Integration Problem Statement If the values of a function f are given at a few points say x0 x1 x n we attempt to estimate a derivative f c oranintegral b a f x dx. The finite elements considered in this paper are those of the Serendipity family of curved isoparametric elements. But isoparametric method application has not Kirchoff theory. Parametric plate element analysis was prefenced Kirchoff theory application. WITH ISOPARAMETRIC QUADRATIC ELEMENTS AND NUMERICAL INTEGRATION O par L. Basic concepts The standard discrete system Finite elements of an elastic continuum displacement approach Generalization of finite element concepts weighted residual and variational approaches. 20 Numerical Integration Num Int 2D CE5610 FINITE ELEMENT ANALYSIS Quiz1 2013 CE5610 FINITE ELEMENT ANALYSIS Chapter 5 Quick return motion CE5610 assign 1 2013 Lecture notes N Siva Prasad CE5610 assign 5 2013 The beginnings of the finite element method actually stem from these early numerical methods and the frustration associated with attempting to use finite difference methods on more difficult geometrically irregular problems. The element is called CST because it has a constant strain throughout it. 7 Numerical Integration for isoparametric T6 and Q8 elements 32 2. C3D8I . WAHLBIN 2 Communiqu par V. Factorization of the volume integral makes the integration independent of the global node coordinates. In this blog post we discuss where and why numerical integration is used. 16. W i is the weight of the function value at that Gauss point. 2 Two simple non conforming plate elements 361 19 Isoparametric finite elements 364 19. 6 Higher Order Isoparametric Quadrilateral Elements 176. 9 6. e IntroFEM 03 Isoparametric 1 elements Introduction to the FE method in geosciences Lecture 3. Finite Elements quot from the early beginning to the very end quot x A x E x b x h g x 0. One prerequisite for numerical integraton is an expression for rate of change of length area or volume with respect to intrinsic coordintes A 3 node one dimensional isoparametric element has nodes at x 1 x 0. Assume that the thickness t is constant and that the coordinates for nodes 1 2 3 and 4 are 2 1 8 1 7 6 and 1 5 respectively. What is the purpose of Isoparametric element It is difficult to represent the curved boundaries by straight edges finite elements. the Isoparametric formulation for simple bar element is explained first and it will lead to a simple mathematical computation. This section focuses on quot Linear Algebra quot in Data Science. 18 Vector Computation in Isoparametric Elements 187 The four noded quadrilateral element Lec 15 ppt pdf Practical Considerations in FEM modeling Lec 16 ppt pdf Convergence analysis of FEM results Lec 17 ppt pdf Higher order elements Lec 18 ppt pdf Isoparametric formulation Lec 19 20 ppt pdf Numerical integration in 2D Lec 21 ppt pdf This thorough yet understandable introduction to the boundary element method presents an attractive alternative to the finite element method. What is another name for the numerical integration used in formulating the k matrix for higher order finite elements How does this relate to the points where stress and strain is computed exactly in an isoparametric element Ok to sketch example 15 points 2. Triangular elements are used for the stress analysis of a plate subjected to in plane loads. We brie y review the classical steps to compute the numerical solution. The Finite Element Methods Notes Pdf FEM Notes Pdf book starts with the topics covering Introduction to Finite Element Method Element shapes Finite Element Analysis PEA FEA Beam elements FEA Two dimessional problem Lagrangian Serenalipity elements Isoparametric formulation Numerical Integration Etc. In the case of straight sided isoparametric triangular elements numerical integration is exact provided a sufficient number of integration points are used. 1 7. ISOFINEL Isoparametric finite element code for elastic analysis of two dimensional bodies eBook NASA National Aeronautics and Space Administration Amazon. points elements. Therefore for the majority of the elements a numerical solution is preferred. Element types triangular rectangular quadrilateral sector curved isoparametric elements and numerical integration. Expected Time Commitment Instructional material equivalent to a one semester credit hour class Universit degli studi di Parma. Jacobian global and element local coordinates. Thus there has been much effort aimed at efficient numerical integration Ref 5 6 and 7 . In this method sampling points are located and weight factors are attached to it. 2D Isoparametric Elements and Numerical Integration 2 5 Lecture 2015 2014 f. 633 A UNIT NO. Local Coordinates A local coordinates system whose origin is located within the element in order to simplify the algebraic manipulations in the derivation of Natural co ordinate systems Isoparametric elements Shape functions for iso parametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction to Analysis Software. This method was utulized isoparametric plate shell etc. 14 Elemental Form and Layout 173. 6 Section 7. 1 the finite element codes NONSAP ADINA and STATIC SAP a modified version use this . The Four Node Quadrilateral. tw course_detail. In an is oparametric element the stiffness matrix is obtained by numerical integration with large calculation amount and much computer work time. The integration schemes used for the second order tetrahedra and wedge elements can be found in Stroud 1971 . 807 C 12. Calculated displacements by my program match exactly with The finite element method with isoparametric finite elements is applied to this problem with curved finite elements along the boundary in connection with a numerical quadrature scheme which is used to compute the coefficients of the resulting linear system. 4 6 node subparametric triangul r element 27 2. Divided cross section of a dam for an isoparametric element. It has been observed see e. Stiffness Integration 283. 8 10 Residual form of nite element equations Rest of FEM Chapter 3 Take Home Final Examination 4. a Para symmetric b dia symmetric c Axi symmetric d All Ans c 2 To About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy amp Safety How YouTube works Test new features Press Copyright Contact us Creators isoparametric elements involves expression such as the one given in Eqn e where the elements of the matrices are necessarily rational polynomials Evaluation of these integrals in close forms is very difficult. Tech Finite Element Methods Mid II November 2014 Question Paper As it is usual in non linear isoparametric finite element analysis in the numerical integration the appropriate material moduli are evaluated at the integration points 39 and therefore the free surface passes in general through the elements. numerical integration 30 8 beam elements 32 8. Hereby the authors used 3 x 3 and 4 x 4 numerical integration for the isoparametric element. 10 Degradation of Isoparametric Elements 155. Elements can be sparse in the area with a small stress gradient and should be dense in the area with a large stress gradient. Element Force Vectors 279. These element integrations represent the main computational effort of the BEM. 2 Isoparametric elements IntroFEM 03 Isoparametric elements 2 Motivation Gauss Legendre Quadrature Numerical integration with Gauss Legendre Quadrature only works on an idealized Element For x 1 to 1 in 1D Numerical Integration MCQs Here you will find MCQ Questions related to quot Numerical Integration quot in Finite Element Method These Numerical Integration MCQ Questions Will help you to improve your Finite Element Method knowledge and will prepare you for various Examinations like Competitive Exams Placements Interviews and other Entrance Exmaniations Higher order elements HW7 due 26 Nov 19 Isoparametric formulation 29 Nov NO CLASS Thanksgiving break 3 Dec 20 Isoparametric formulation 6 Dec 21 Numerical Integration in 2D HW8 due 10 Dec QUIZ 2 Lec 12 21 Project due The isoparametric shape functions for CST element are N1 N2 and N3 1 . The unloading was not taken into account in Part 1 of this paper however it must This makes our integration easy. Four Node Quadrilateral for Axisymmetric Problems. Chapter 10 Isoparametric Formulation and Numerical Integration 10. 811 D 14. Finite elements with this geometry are extensively used in modeling three dimensional solids. Main reason for its popularity is that the method results in computer codes which are versatile in nature that can solve many practical problems with minimum training. 2 A REVIEW OF XFEM DEVELOPMENT 61 3. 1 Independence of the Basis 5. Finite elements IV 2D boundary value problems. unit v isoparametric formulation Natural co ordinate systems Isoparametric elements Shape functions for iso parametric elements One and two dimensions Serendipity elements Numerical integration and application to plane stress problems Matrix solution techniques Solutions Techniques to Dynamic problems Introduction The finite element solution of elliptic and parabolic equations using simplicial isoparametric elements Nedoma Josef ESAIM Mathematical Modelling and Numerical Analysis Mod lisation Math matique et Analyse Num rique Tome 13 1979 no. 7. Motivation In the recent years un tted nite element methods have drawn more and more attention. contribution to the field of finite element analysis during the past 40 years. Isoparametric elements are finite elements for which the approximation of geometry is performed using the 9. Numerical integration of stiffness matrices in finite element analysis can represent a significant portion of the overall computational effort. Notes FE 2D time dependent solution Problem set 2 D FE heat equation NURBS Enhanced Finite Element Method NEFEM allows a seamless integration of the CAD boundary representation of the domain and the finite element method. For instance a stiffness elemental matrix coef cient is computed as Ke ij h e xNi n x xNi n x d I J 1 NUMERICAL INTEGRATION REVIEW Note that the exact integration is obtained if the integrand y x is a polynomial of order 2 NG 1 or less. integration by parts boundary conditions and coordinate mappings . 4 Approximation of Domain and Boundary Conditions 5. 3 cantilever beam problem 38 9 isoparametric mapping in two dimensions 42 10 a four node plane stress element 45 10. i Evaluate x2 xy2 dx dy by Gauss numerical integration. Numerical integration is carried out using Gauss quadrature points with the advantage over triangular elements that the integration is carried out from 1 to 1 in orthogonal directions. We validate this approach by introducing the new quadratureformulas into a multiscale nite element method for the 2D reaction di usion equation. In this paper the same integrations are performed analytically ngsxfem is an add on library to Netgen NGSolve a general purpose high performance finite element library for the numerical solution of partial differential equations. For the second order elements Ga uss integration is always used because it is efficient and it is especially suited to the polynomial product interpolations used in these elements. See also 13 for the application of Commonly available numerical integration methods can be used. 87 No. 6 8 node rectangular element 31 2. au Kindle Store Note It is not correct say isoparametric element because an element is unique in terms of the number of nodes and degrees freedom per node hence the approximation order on the element is unique. Finite elements with constant Jacobian in the isoparametric map so that the nodal shape functions and their derivatives are polynomials on the physical element are suitable for the HNI method. quot The interpolating functions are shown below formulation of isoparametric elements along with the aspect of numerical integration that is essential for computations with isoparametric elements is considered in the next chapter. 1 bernoulli beam 32 8. Implementation of a finite element computer code Numerical methods for PDEs FEM implementation element stiffness matrix isoparametric mapping assembling global stiffness Numerical integration The expressions above are for a general higher order element known as an isoparametric element where the sides of the element need not be straight. NASTRAN CARDS COMMON TO ALL ANALYSES NASTRAN was originally developed by NASA for the space program at the end of the 1960 s. 6 Chapter 10 HW 6 6. 4 A plane truss has two bars. Week 10 Quadrilateral elements The isoparametric formulation Shape functions Numerical Integration. elements that ensure the continuity of the displacement field across boundaries How . Solution of 1D problems with quadratic elements. 6 Calculation of element stiffness 94 3. 67. The item Numerical methods in finite element analysis Klaus J rgen Bathe Edward L. Meshing using triangles. In the axisymmetric models a mean radius approximation is often used in which the BTDBr product is evaluated at the element s centroid prior to integration 1 6 9 . It considers an exact representation of the geometry by means of the usual CAD description of the boundary with Non Uniform Rational B Splines NURBS . g. Variational Crimes 4. For elements not intersecting the boundary a standard finite element interpolation and numerical integration are used preserving the efficiency of the classical finite element method. The functions to be integrated over elements can be quite complex and therefore require numerical treatment . A. 55 Why numerical integration is required for evaluation of stiffness matrix of an Isoparametric element AU Nov Dec 2011 5. The nodal coordinates of interior Point P are node 1 2 2 node 2 5 3 and node 3 3 10 . Gauss quadrature is a means for numerical integration which evaluates an integral as the sum of a finite number of terms where i is the value of at i . The basis of NEFEM in 2D are presented in 12 showing the advantages in front of classical isoparametric FEs using both continuous and discontinuous Galerkin formulations. 15 Inconsistent Elements 176. 3 Eigenvalue and Eigenfunction mesh the choice of the nite element degrees of freedom nite element space etc and the quadrature rules used in the next sections. 1D Isoparametric Elements 2 4 Lecture 2015 e. 2 timoshenko beam 35 8. 1 Independence of the Basis 205 5. The three dimensional local average is more compre hensive and the most suitable for the actual situation. Sympos. Saouma David Schwemmer Numerical evaluation of the quarter point crack tip element International Journal for Numerical Methods in Engineering 10. Numerical integrations modeling considerations Practical use of numerical integration. In almost all c. 4 Higher Order isoparametric master element as shown in Figure 2. 3 Eight node isoparametric quadrilateral element 370 19. 5 Numerical Integration for Isoparametric Quadrilateral Elements 165. Accordingly a transformation matrix J called Jacobian is produced. introducing most of the nite element toolbox but with postponing demanding topics related to di erential equations e. Stress Calculations 284. Background Holistic Numerical Methods licensed under a Creative Commons Attribution NonCommercial NoDerivs 3. Conjugate Gradient Implementation of the Quadrilateral Element. Veryard 7 Irons and Razzaque 5 Barlow 1 that applying quadratic members of this family The numerical simulations of slope stability problems have been performed by using the triangle element 20 23 and the isoparametric element 16 19 . If these elements satisfy the convergence requirements of constant stress representation and strain free rigid body motions for one geometry then it will satisfy the conditions for any geometry. Discretization of functions in 2D with linear triangular elements. Numerical experiments for 2D propagation problems demonstrate the applicability and benefits of the isoparametric technique for simulations involving curved domains Most nite element FE software uses numerical integration to compute FE stiffness matrices. 5 Numerical integration 91 3. Some element stiffness matrices are easily developed in closed form. Solution of 2 D heat equation. Finite element applications to linear elliptic boundary value problems in structures solid and fluid mechanics and heat transfer. 1620200907 20 9 1629 1641 2005 . boundary condition problems ill conditioned problems FE model improving elastic supports 6 hours . 1 for beam with cubic shape functions if the exact solution is a polynomial of order quartic 2 for 8 node isoparametric plane elements what many of you call QUAD8 right if the exact solution is a cubic and 3 for 20 node isoparametric solid elements if the exact solution is a cubic. 6 Higher Order Versus Lower Order Elements References 5 THE PATCH TEST 5. 2D element edge definition. umbering scheme for their three dimensional isoparametric element. v T the displacement elements can be represented as u N 1 q 1 N 2 q 3 N 3 q 5 N 4 q 7 v N 1 q 2 N 2 q 4 N 3 q 6 N 4 q 8 Next we will introduce numerical integration methods for evaluating the quadrilateral element stiffness matrix. The paper describes Numerical integration refers to all the procedures algorithms and techniques in the numerical analysis to obtain an approximate solution to a definite integral. This representation is illustrated on speci c elements. AU Nov Dec 2016 5. step numerical integration procedure. 3. The add on enables the use of geometrically unfitted finite element technologies known under different labels e. Sevilla S. Content of the Master Studies Continuum Mechanics I numerical integration FEM 3. Wiley amp Sons. 4 Loads Masses and Output 4. The mapping from x y coordinate system to s t coordinate is given by the shape functions written in terms of parent element. The use of isoparametric elements and numerical integration dramatically increases the robustness of the finite element analysis method. AS2009 Thursday 10 12 NO D 11 The value of integration of x e x over the limits 0. Key words Numerical integration Finite Element Method 1 Introduction Finite element methods are highly popular because among other reasons they are good and 2. JAN BACKLUND Department of Mechanical Engineering Linkoping Institute of Technology Linkoping Sweden In a recent paper Stricklin et all have compared the eight node isoparametric element with f quadrilateral elements consisting o two or four six node linear strain triangles. THOM E Abstract. AC bar is steel and its cross section area is 100 mm2. Reduced integration elements do not permit internal stress gradients because of their single integration point and suffer from their own numerical problem known as. This set of Finite Element Method Multiple Choice Questions amp Answers MCQs focuses on Two Dimensional Isoparametric Elements Four Node Quadrilateral . 55. 20 2 Then we can determine the number of appropriate integration points by investigating the maximum order of BTCB. the boundary of the element is replaced by a numerical ux Fb ne Ue U out e which is evaluated in terms of the solution in the current element e and the solution at neighboring elements Uout e x lim 0 U x nee. 6 Isoparametric transformation numerical integration quads. 5 Hierarchical Elements 4. Two Dimensional Integrals 283. Optimality convergence function spaces and energy norms. That is for a given discretization with polynomials of order p a numerical quadrature with p 2 Gauss Legendre integration points in each patch provides the maximum accuracy. 8 15 16 17 19 27 . Univ. 2 Integration rules in triangular domains for q 1 left q 2 center and q 3 right . Boundary loads are supported and computed using numerical integration. Most finite element FE software uses numerical integration to compute FE stiffness matrices. 2. SO in this formulation we are reducing the length of dimensions of the coordinates for simplyfing our integration. 6. Most of the engineering structure is not regular in shape and some of them even have very complicated boundary shapes. 1 Introduction 273. 7 6. Two Dimensional Isoparametric Elements and Numerical Integration. Instead stiffness matrices and load vectors are typically evaluated numerically using Gauss quadrature rule over triangular or quadrilateral regions. Matrix assembly. Hence they should be considered during the integration of adjacent as well. assembly of the element mass matrix and stiff matrix can be carried out by conventional finite element method. By using Lagrangian method the equations of motion for two link flexible manipulator were obtained. q q MCq. Stefan Schmalholz ETH Zurich Numerical modeling of rock deformation 10 FEM 2D 9 node element numerical integration and isoparametric elements Stefan Schmalholz schmalholz erdw. 15 Inconsistent Elements 176 8. 4 ISOPARAMETRIC ELEMENTS AND NUMERICAL INTRGRATION 1 The two dimensional elements are called elements. 6. Computers amp Mathematics with Applications 68 12 1945 1962. Three inde pendent isoparametric coordinates are used to map the six noded triangular element as shown in Figure 3. Reduced integration linear isoparametric elements use uniformly reduced integration. 1 Numerical Integrations Numerical integration is an important part of the finite element technique. 1 displacements and geometry of four node plane element 47 1. pdf from MANE 4240 at Rensselaer Polytechnic Institute. All the isoparametric solid elements are integrated numerically. After integration is done the length of dimesnions should be replaced it is done by the Jacobian. This is not a limitation per se since the X FEM permits complex crack geometries to be This is a discussion of some numerical integration methods for the surface integral AQ do 1. to employ a 8. For its numerical solution we employ the finite element method with quadratic triangular isoparametric l ments combined with a numerical int gration procedure involving function values at three nodal points in each element. The discretized methods make use of numerical differentiation and numerical integration. Numerical integration in the computational element h e approximation of e is performed using the isoparametric transformation given in Equation 2 with a numerical quadrature in the reference element I. Natural Coordinates Triangular Elements Rectangular Elements Lagrange and Serendipity Elements Solid Elements Isoparametric Formulation Stiffness Matrix of Isoparametric Elements Numerical Integration One Dimensional Numerical Integration Two and Three Dimensional Analysis of Frame Structures. This is due to the fact that with these interpolations the transverse shear strains cannot vanish at all points in the element when it is subjected to a constant bending moment. essential elements of all the numerical methods currently used extensively in Numerical Integration 126 3. Create a free account to download. The approach taken is mathematical in nature with a strong focus on the Isoparametric finite elements natural coordinates Jacobian Gauss integration method 10 hours . 1 INTRODUCTION 61 3. International Centre for Mechanical Sciences Courses and Lectures vol 130. Also The expressions above are for a general higher order element known as an isoparametric element where the sides of the element need not be straight. Numerical Integration. 5 Timoshenko Beam Element 269 linear element. pdf or read online for free. isoparametric elements and numerical integration mcq