THE FINITE ELEMENT METHOD for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 3: THE FINITE ELEMENT METHOD

CONTENTS STRONG AND WEAK FORMS OF GOVERNING EQUATIONS HAMILTON’S PRINCIPLE FEM PROCEDURE Domain discretization Displacement interpolation Formation of FE equation in local coordinate system Coordinate transformation Assembly of FE equations Imposition of displacement constraints Solving the FE equations STATIC ANALYSIS EIGENVALUE ANALYSIS TRANSIENT ANALYSIS REMARKS

STRONG AND WEAK FORMS OF GOVERNING EQUATIONS System equations: strong form, difficult to solve. Weak form: requires weaker continuity on the dependent variables (u, v, w in this case). Weak form is often preferred for obtaining an approximated solution. Formulation based on a weak form leads to a set of algebraic system equations – FEM. FEM can be applied for practical problems with complex geometry and boundary conditions.

HAMILTON’S PRINCIPLE “Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.” An admissible displacement must satisfy: The compatibility equations The essential or the kinematic boundary conditions The conditions at initial (t1) and final time (t2)

HAMILTON’S PRINCIPLE Mathematically where L=T-P+Wf (Kinetic energy) (Potential energy) (Work done by external forces)

FEM PROCEDURE Step 1: Domain discretization Step 2: Displacement interpolation Step 3: Formation of FE equation in local coordinate system Step 4: Coordinate transformation Step 5: Assembly of FE equations Step 6: Imposition of displacement constraints Step 7: Solving the FE equations

Step 1: Domain discretization The solid body is divided into Ne elements with proper connectivity – compatibility. All the elements form the entire domain of the problem without any overlapping – compatibility. There can be different types of element with different number of nodes. The density of the mesh depends upon the accuracy requirement of the analysis. The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.

Step 2: Displacement interpolation Bases on local coordinate system, the displacement within element is interpolated using nodal displacements.

Step 2: Displacement interpolation N is a matrix of shape functions Shape function for each displacement component at a node where

Displacement interpolation Constructing shape functions Consider constructing shape function for a single displacement component Approximate in the form pT(x)={1, x, x2, x3, x4,..., xp} (1D)

Pascal triangle of monomials: 2D

Pascal pyramid of monomials : 3D

Displacement interpolation Enforce approximation to be equal to the nodal displacements at the nodes di = pT(xi) i = 1, 2, 3, …,nd or de=P  where ,

Displacement interpolation The coefficients in  can be found by Therefore, uh(x) = N( x) de

Displacement interpolation Sufficient requirements for FEM shape functions (Delta function property) 1. (Partition of unity property – rigid body movement) 2. 3. (Linear field reproduction property)

Step 3: Formation of FE equations in local coordinates Since U= Nde Strain matrix e = LU Therefore,  e = L N de= B de e T Ve V c Π d B Bd ) ( 2 1 ε ò = V c T Ve e d B k ò = or where (Stiffness matrix)

Step 3: Formation of FE equations in local coordinates Since U= Nde  or where (Mass matrix)

Step 3: Formation of FE equations in local coordinates (Force vector)

Step 3: Formation of FE equations in local coordinates (Hamilton’s principle)  FE Equation

Step 4: Coordinate transformation x y x' y' Local coordinate systems Global coordinate systems (Local) (Global) where , ,

Step 5: Assembly of FE equations Direct assembly method Adding up contributions made by elements sharing the node (Static)

Step 6: Impose displacement constraints No constraints  rigid body movement (meaningless for static analysis) Remove rows and columns corresponding to the degrees of freedom being constrained K is semi-positive definite

Step 7: Solve the FE equations for the displacement at the nodes, D The strain and stress can be retrieved by using e = LU and s = c e with the interpolation, U=Nd

STATIC ANALYSIS Solve KD=F for D Gauss elmination LU decomposition Etc.

EIGENVALUE ANALYSIS [ K - li M ] fi = 0 (Homogeneous equation, F = 0) Assume Let  (Roots of equation are the eigenvalues) [ K - li M ] fi = 0 (Eigenvector)

EIGENVALUE ANALYSIS Methods of solving eigenvalue equation Jacobi’s method Given’s method and Householder’s method The bisection method (Sturm sequences) Inverse iteration QR method Subspace iteration Lanczos’ method

TRANSIENT ANALYSIS Structure systems are very often subjected to transient excitation. A transient excitation is a highly dynamic time dependent force exerted on the structure, such as earthquake, impact, and shocks. The discrete governing equation system usually requires a different solver from that of eigenvalue analysis. The widely used method is the so-called direct integration method.

TRANSIENT ANALYSIS The direct integration method is basically using the finite difference method for time stepping. There are mainly two types of direct integration method; one is implicit and the other is explicit. Implicit method (e.g. Newmark’s method) is more efficient for relatively slow phenomena Explicit method (e.g. central differencing method) is more efficient for very fast phenomena, such as impact and explosion.

Newmark’s method (Implicit) Assume that Substitute into

Newmark’s method (Implicit) where Therefore,

Newmark’s method (Implicit) Start with D0 and Obtain using March forward in time Obtain using Obtain Dt and using

Central difference method (explicit) (Lumped mass – no need to solve matrix equation)

Central difference method (explicit) Find average velocity at time t = -t/2 using Find using the average acceleration at time t = 0. Find Dt using the average velocity at time t =t/2 Obtain D-t using D0 and are prescribed and can be obtained from Use to obtain assuming . Obtain using Time marching in half the time step Central difference method (explicit)

REMARKS In FEM, the displacement field U is expressed by displacements at nodes using shape functions N defined over elements. The strain matrix B is the key in developing the stiffness matrix. To develop FE equations for different types of structure components, all that is needed to do is define the shape function and then establish the strain matrix B. The rest of the procedure is very much the same for all types of elements.