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

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Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS STRONG AND WEAK FORMS OF GOVERNING EQUATIONS HAMILTONS 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

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Finite Element Method by G. R. Liu and S. S. Quek 3 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.

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Finite Element Method by G. R. Liu and S. S. Quek 4 HAMILTONS 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 (t 1 ) and final time (t 2 )

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Finite Element Method by G. R. Liu and S. S. Quek 5 HAMILTONS PRINCIPLE Mathematically where L=T +W f (Kinetic energy) (Potential energy) (Work done by external forces)

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Finite Element Method by G. R. Liu and S. S. Quek 6 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

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Finite Element Method by G. R. Liu and S. S. Quek 7 Step 1: Domain discretization The solid body is divided into N e 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.

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Finite Element Method by G. R. Liu and S. S. Quek 8 Step 2: Displacement interpolation Bases on local coordinate system, the displacement within element is interpolated using nodal displacements.

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Finite Element Method by G. R. Liu and S. S. Quek 9 Step 2: Displacement interpolation N is a matrix of shape functions where Shape function for each displacement component at a node

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Finite Element Method by G. R. Liu and S. S. Quek 10 Displacement interpolation Constructing shape functions – Consider constructing shape function for a single displacement component – Approximate in the form p T (x)={1, x, x 2, x 3, x 4,..., x p } (1D)

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Finite Element Method by G. R. Liu and S. S. Quek 11 Pascal triangle of monomials: 2D

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Finite Element Method by G. R. Liu and S. S. Quek 12 Pascal pyramid of monomials : 3D

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Finite Element Method by G. R. Liu and S. S. Quek 13 Displacement interpolation – Enforce approximation to be equal to the nodal displacements at the nodes d i = p T (x i ) i = 1, 2, 3, …,n d or d e =P where,

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Finite Element Method by G. R. Liu and S. S. Quek 14 Displacement interpolation – The coefficients in can be found by – Therefore, u h (x) = N( x) d e

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Finite Element Method by G. R. Liu and S. S. Quek 15 Displacement interpolation Sufficient requirements for FEM shape functions 1. (Delta function property) 2. (Partition of unity property – rigid body movement) 3. (Linear field reproduction property)

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Finite Element Method by G. R. Liu and S. S. Quek 16 Step 3: Formation of FE equations in local coordinates Since U= Nd e Therefore, = LU = L N d e = B d e Strain matrix orwhere (Stiffness matrix) e T VeVe T ee TT e VeVe T VeVe VcVcVcΠddBBddBdBdd)( εε 2 1 Vc T VeVe e dBBk

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Finite Element Method by G. R. Liu and S. S. Quek 17 Step 3: Formation of FE equations in local coordinates Since U= Nd e orwhere (Mass matrix)

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Finite Element Method by G. R. Liu and S. S. Quek 18 Step 3: Formation of FE equations in local coordinates (Force vector)

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Finite Element Method by G. R. Liu and S. S. Quek 19 Step 3: Formation of FE equations in local coordinates FE Equation (Hamiltons principle)

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Finite Element Method by G. R. Liu and S. S. Quek 20 Step 4: Coordinate transformation x y x'x' y'y' y'y' x'x' Local coordinate systems Global coordinate systems,, where (Local) (Global)

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Finite Element Method by G. R. Liu and S. S. Quek 21 Step 5: Assembly of FE equations Direct assembly method – Adding up contributions made by elements sharing the node (Static)

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Finite Element Method by G. R. Liu and S. S. Quek 22 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

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Finite Element Method by G. R. Liu and S. S. Quek 23 Step 7: Solve the FE equations Solve the FE equation, for the displacement at the nodes, D The strain and stress can be retrieved by using = LU and = c with the interpolation, U=Nd

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Finite Element Method by G. R. Liu and S. S. Quek 24 STATIC ANALYSIS Solve KD=F for D – Gauss elmination – LU decomposition – Etc.

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Finite Element Method by G. R. Liu and S. S. Quek 25 EIGENVALUE ANALYSIS (Homogeneous equation, F = 0) Assume Let [ K i M ] i = 0 (Eigenvector) (Roots of equation are the eigenvalues)

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Finite Element Method by G. R. Liu and S. S. Quek 26 EIGENVALUE ANALYSIS Methods of solving eigenvalue equation – Jacobis method – Givens method and Householders method – The bisection method (Sturm sequences) – Inverse iteration – QR method – Subspace iteration – Lanczos method

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Finite Element Method by G. R. Liu and S. S. Quek 27 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.

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Finite Element Method by G. R. Liu and S. S. Quek 28 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. Newmarks 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.

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Finite Element Method by G. R. Liu and S. S. Quek 29 Newmarks method (Implicit) Assume that Substitute into

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Finite Element Method by G. R. Liu and S. S. Quek 30 Newmarks method (Implicit) where Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 31 Newmarks method (Implicit) Start with D 0 and Obtainusing Obtain using Obtain D t andusing March forward in time

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Finite Element Method by G. R. Liu and S. S. Quek 32 Central difference method (explicit) (Lumped mass – no need to solve matrix equation)

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Finite Element Method by G. R. Liu and S. S. Quek 33 Central difference method (explicit) D, t x x x x x t0t0 t- t- t/2 t/2 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 D 0 and are prescribed and can be obtained from Use to obtain assuming. Obtain using Time marching in half the time step

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Finite Element Method by G. R. Liu and S. S. Quek 34 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.

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