Formulation of the displacement-based Finite Element Method and General Convergence Results
z Basics of Elasticity Theory strain e: measure of relative distortions u r r' y for small displacements : x r : before deformation r' : after deformation u : displacement compatibility of the strain field: d/dx u e int. + comp.
Basics of Elasticity Theory deformation creates internal forces C: material matrix (symmetric) e : strain ti : stress b.c. deformed solid solid in equilibrium the problem of elasticity: find u in the presence of boundary conditions: (essential) (natural)
The principle of virtual displacements for any U that obey the b.c. and the true equilibrium stress field: internal virtual work virtual work by external forces stress field in equilibrium with virt. strain derived from virt. displacements U assume true U U is the solution implicitly contains: e force boundary conditions direct stiffness method (in FEM formulation) no yes t p. o. v. d. satisfied for any U?
FEM: domain discretization global, nodal displacements: node displ. in global coord. interpolation matrix H(m): strain-displacement matrix B(m): it holds: global coord. system local coord. system of m-th element solid decomposed into assemly of finite elements: no overlap no holes nodal points coincide principle of virtual displacements for the discretized body:
FEM: matrix equations ansatz for displacement field discretized principle of virt. displ. ansatz for virt. displacement field solution when p.o.v.d. fullfilled with one obtains: implicit direct stiffness especially: = ei, where boundary forces contained discrete displacement field to be determined
FEM: matrix equations dynamic case with dissipative forces ~ du/dt: modified body forces: define: inertial forces damping forces K and M are symmetric
FEM: Two-element bar assemblage two 2-point elements, polynomial ansatz u(m)(x) linear: interpolation condition: 1 m=1
FEM: Two-element bar assemblage calculate B(1)(x) for given u(1)(x): m = 1: analogous: material matrix:
FEM: Two-element bar assemblage slow load application:, RS= RI=0 obtain U(t*) by solving:
FEM: Exact stiffness vs. FEM approx. from variational principle: A=(1+x/40)2 exact solution w. b.c. u(0) = 0 and u(80) = 1: forces at the ends of the bar: FEM (exact stiffness matrix) displacement ansatz too rigid (FEM approx., prev. example) overestimate of stiffness
FEM: Displacement constraints decompose U = FEM equations: : unconstrained : constrained modified force vector if constraints don't coincide with Ub, try: where Ub can be constrained need to transform FEM matrices accordingly:
Refining the FEM solution keep element size keep order of interpolation p refinement h/p refinement: both simultaneously h refinement
Convergence of a FEM solution strain energy pos. def. bilinear form Convergence in the norm induced by a: measures discretization and interpolation errors, only monotonic convergence: a(uh,uh) necessary: a(u,u) compatibility of mesh and elements completeness of elements 1/h
mon. Conv.: Compatibility Compatibility: continuity of u within and across element boundaries discontinuity different interpolation is adjacent elements Example: discontinuity calculation of a(uh,uh): u d/dx e e C t delta function gives unphysical contributions to a (uh,uh) and destroys monotonic convergence different element size in adjacent elements
mon. Conv.: Completeness Completeness: element must be able to represent all rigid-body modes and states of constant strain Reason: Number of rigid-body modes: 1,...ln #rbm = dim(ker(k)) rbm determined by the eigenvectors of K:
Convergence rates Assumptions: interpolation w. complete polynomials up to order k Lack of smoothness slower convergence non-uniform grid: h refinement: exact solution is smooth enough, such that Sobolev-Norm of order k+1 is finite: h/p refinement: uniform mesh Estimate (p-refinement): Def.: (Sobolev-Norm of order 0)
Error assessment FEM solution violates differential equilibrium: non-continous stress possible Graphical assement: Isoband-plots
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