Seoul National UniversityAerospace Structures Laboratory Seoul National University Chapter 6 EXTENSIONS Introduction to Finite Element Method

Seoul National UniversityAerospace Structures Laboratory 6.1 Introduction We have now reached a point where most of the ideas essential to an introduction to finite element methods have been covered. There are, however, several remaining concepts not dealt with earlier which deserve some attention in a first course on the subject. These are natural extensions of previous result to: 1.Three-dimensional problems. 2. Fourth-order boundary-values problems. 3. Systems of differential equations (such as those encountered in problems in which the solution is vector-valued). 4. Time-dependent problems. (we already dealt this in Chapter 3) Our purpose here is to give a brief account of extension of idea discussed earlier to each of the three subjects listed above. We concentrate on only the formulative aspects of the method as it applies to these problems, leaving the discussion of details and limitations to higher level courses. 6.1 INTRODUCTION

Seoul National UniversityAerospace Structures Laboratory Find such that 6.2 Three Dimensional Problems with Jump condition Boundary Condition 6.2 THREE DIMENSIONAL PROBLEMS

Seoul National UniversityAerospace Structures Laboratory Find such that And then we have the following Variational Formulation Let us apply the Galerkin Approximation with This will, of course, end up the linear system, 6.2 THREE DIMENSIONAL PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 1.We regard the mesh as being described by a sequence of transformations from a fixed master element onto each finite element 2.The local shape functions are images of polynomial shape functions defined over, and nodal points are located so that the final global basis functions are continuous across interelement boundaries 3.If is a typical test function in, then is in and, since 6.2 THREE DIMENSIONAL PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 6.2 THREE DIMENSIONAL PROBLEMS The element calculations are performed with numerical quadrature on the master element

Seoul National UniversityAerospace Structures Laboratory 6.3 FOURTH ORDER PROBLEMS Two-point Boundary-Value Problems B.C D.E

Seoul National UniversityAerospace Structures Laboratory VF 1. The coefficients in B.C must be such that 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory In which case we solve B.C to obtain Wherein th constants are determined by the coefficients in B.C; for example, 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 2. If we wish to identify a class of admissible functions on which smoothness requirements are barely strong enough to make this integral well defined, it is sufficient to take and to be members of a class of functions, denoted, whose derivatives of order 2 and less are square-integrable over. In other words, a test function will belong to if 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 3. We see that the variational statement for our model fourth-order boundary- value problem takes the following form : find such that 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory We have essential boundary condition of the form Then the (6.3.7) problem reduces to on of finding which satisfies (6.3.8) and Wherein denotes the class of functions satisfying (6.3.6) and which have the property that 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory Finite Element Approximations The construction of a finite element approximation of fourth order problems, as usual, based on Galerkin’s method. Find such that Where is an approximate finite-dimensional subspace of 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory We outline the essential features of the method as follows : 1.The finite element basis function must be such that their first derivatives are continuous throughout the domain of the approximate solution 2.The test function assume the form where N is the number of nodes in the mesh and the basis function have the properties 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 3. We can construct cubic shape function which exhibit the necesary properties : These conditions uniquely define the local cubic Hermite polynomial shape functions, 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory A two-dimensional problem The biharmonic equation can be written by With various boundary conditions VF is 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 16 D.O.F tensor product of 1-D Hermite polynomial We need the shape functions satisfying the condition in two-dimension; ex) thin plate theory (Kirchhoff) thick Reissner-Mindlin plates i) 4 node C 1 rectangular element 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory ii) Quintic Triangle 21 D.O.F Too many D.O.F Incompatible element Hybrid or Mixed formulation Stability Patch Test 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 6.3 FOURTH ORDER PROBLEMS

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS A one-dimensional problem or system of equations : more than two unknowns per each node For convenience, we assume homogeneous Dirichlet data at the ends, Letting and be test functions, we form the weighted-residual statement for each individual equation int (6.4.1). For, and sufficiently smooth, we have

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS Integrating by parts and setting at the ends and, we obtain the desired variational boundary-value problem: find and such that for all test functions and The representations of functions and in are of the form The finite element problem is to find and such that

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS Introducing representations of the form (6.4.5) into (6.4.6) leads to the finite element system

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS Plane-Stress Problems In elasticity, the state variable is the displacement vector and the flux is the stress vector. Here are the components of stress and are components of strain Equilibrium equations are Then, Navier- equation

Seoul National UniversityAerospace Structures Laboratory a) plane stress a) is obtained by b) b) 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS As our example, we study the plane-stress problem for an isotropic linearly elastic body. The governing equations can be written in concise form by introducing the matrices

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS The conservation law for this problem, which expresses the principle of conservation of linear momentum for the body (static equilibrium in this case), now assumes the form If and are sufficiently smooth, this leads to the system of PDE of equilibrium for the body: The constitutive equation characterizing the material as linearly elastic, homogeneous, and isotropic is simply Where the strain are related to the state variable according to the strain-displace -ment relations,

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS Introducing (6.4.11) and (6.4.12) into (6.4.10) gives the governing system of PDE eqns in terms of the displacement vector : The conservation principle, applied to the portion of, yields the condition Whereas the specification of the displacement vector on the complementary portion of the boundary is characterized by the condition If is an arbitrary admissible displacement vector, then we wish to find such that, and

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS For a typical element such as that shown in Fig 6.6, we have where and

Seoul National UniversityAerospace Structures Laboratory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS The equations governing a typical element are where

Seoul National UniversityAerospace Structures Laboratory One-dimensional Euler beam theory : 4 th order problem 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS

Seoul National UniversityAerospace Structures Laboratory Timoshenko beam theory 6.4 SYSTEMS OF DIFFERENTIAL EQUATIONS

Seoul National UniversityAerospace Structures Laboratory Mixed & Hybrid method – variables other than Displacement (irreducible of mixed) Necessity 1.constraint to variational formulation for example, incompressible condition, Lagrange Multiplier for this condition multiplier → pressure 2.complex variables in physical or mechanical formulation 3.difficult to satisfy high order continuity (H 2 ) at 4th order problem, etc. 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory a) Constrained minimization then, other method ⇒ Lagrange multiplier, q then, and variational formulation ; 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory Ex 1) from boundary condition, Dirichlet condition, then, Variational formulation ; 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory Ex 2) Stoke's problem in 2D (slow Incompressible flow) variational formulation ; 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory b) Mixed FE Approximation 1) 2) other mixed method or 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory ex) 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory c) Hybrid methods conformining element keep continuity for variational form. In some particular cases, weak the continuity and discretize, impose the continuity to constraint condition as Lagrange multiplier (satisfaction in average sense) 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory d) Independent discretization 6.5 MIXED & HYBRID FORMULATION original decomposition

Seoul National UniversityAerospace Structures Laboratory Penalty method, Reduced Integration Penalty method  approximation of Lagrange multiplier (Perturbed Lagrangian) 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory then, Locking but sometimes overconstrained, locked Reduced or Selective Integration 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory The Patch Test ◎ Irons.B.M & Razzaque " Experience with the patch test for convergence of finite elements" Math. Found. of FEM with appl to PDE A.K Azis ◎ Irons & Razzaque " Shape Function formulations for elements other than displacement models" Variational Methods in Engineering 1973 ◎ Irons B.M The Patch Test for engineers Proc. FE synp But ratio test → convergence But 100 digit computer 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory ◎ Procedure ⅰ ) when arbitrary shape patch is selected, at least 6 digit exact solution under constant stress could be obtained? ⅱ ) impose any specific loading equivalent to constant stress to prevent Spurious success ⅲ ) impose boundary nodal values equivalent to the assumed Stress a) impose displacement for all Nodes b) calculate nodal force by traction force c) impose displacement to boundary node ◎ Hourglassing, Chicken net, Spurious mode 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory ◎ Patch Test Irons convergence test of FE by numerical method - Node existing inside the mesh - Interelement side existing - Non-uniform element size In these conditions, FE solutions should be almost same with exact solutions under loading of constant stress 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory Weak Patch Test: Refine → pass Element Shape, Connection, and grading Large aspect ratio Near triangle off center node Highly Skewed Triangular Quadralatural Cuved Side 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory * Grading Problem * Connection 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory where, if 6 node quadratic triangles and u is smooth ◎ Diffusion and chemical reaction of species ◎ Neutron flux (Transport and diffusion equations) 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory ◎ Ideal MHD Plasma Equilibria ◎ Eletric and Magnetic field ◎ Vibration of wind tunnel velocity potential term, 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory at Solid wall at Slot given eq. ; and 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory 2-D Applications (i) Mesh numbering - Band optimization (ii) Front width optimization (iii) h-p convergence (iv) Automatic mesh generation method +FEM (v) Substructuring (vi) Concurrent or Parallel computing method (vii) Condensation (static, dynamic) effect Time, Solution accuracy (viii) Mixed Problem 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory *FEM modeling and Preprocessing Beam Plate & shell best model Pipe 3D element accurate solution at minimum time and cost Proper element selection is important Element pattern change by Transition consider flow analysis mismatch consider Temperature change Singularity Aspect ratio 3 : 1 for stress 10 : 1 for displacement integration rate of function used 6.5 MIXED & HYBRID FORMULATION

Seoul National UniversityAerospace Structures Laboratory Nonlinearity Material Large deformation Rubber elasticity Visco-Elasto-Viscoplasticity 6.6 NONLINEAR PROBLEMS

Seoul National UniversityAerospace Structures Laboratory * Condensation Static condensation (No approximation) substitute, 6.3 FOURTH ORDER PROBLEMS