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Generalized Finite Element Methods

Generalized Finite Element Methods. Approximation (Partition of Unity). Suvranu De. Discussion on MLS type approximations. Advantages Provides “ local ” shape functions (banded matrices) without having to use a “mesh” ( meshfree approximation )

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Generalized Finite Element Methods

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  1. Generalized Finite Element Methods Approximation (Partition of Unity) Suvranu De

  2. Discussion on MLS type approximations • Advantages • Provides “local” shape functions (banded matrices) without having to use a “mesh” (meshfreeapproximation) • Arbitrary continuity of the approximation spaces • Arbitrary polynomialconsistency of approximation • Ability to enrichapproximation space with (known) asymptotic solution of the governing differential equations (e.g., crack)

  3. Solvability: The matrix A(x) is invertible iff • P is full rank, and • at least m+1 weights are nonzero at x Discussion on MLS type approximations • Disadvantages • Very stringentcondition on the invertibilityof the A(x) matrix (problems in 2/3D) • Functions arenot easy to differentiate and integrate • Once the local basis is chosen, it cannot be changed from one node to the other. • Method is more flexible than FEM, but could be better!

  4. Partition of Unity (PU) Method Has all the advantages of MLS type approximations, but NONE of the disadvantages. Step 1.Define a partition of unity (PU) on the domainon an arbitrary set of nodes. Make sure that the domain is covered by the union of the supports of the PU functions Examples of PU: W1(x) W2(x) 1. Shepard functions W0(x) Wn(x) x2 x0 x1 xn x

  5. Partition of Unity (PU) Method 2. The “hat” functions used in FEM W0(x) x2 x0 x1 xn x The PU property ensures What can we do to ensure that the approximation reproduces, e.g., x on the domain?

  6. Partition of Unity (PU) Method Let us look at the functions x2 x0 x1 xn x Hence we needmultiple shape functions at a node! Step 2.Define a local approximation space at each node

  7. Partition of Unity (PU) Method Step 3.Define the global approximation space as the blend of the local approximation spaces through the PU functions Hence, the PU approximation is

  8. Partition of Unity (PU) Method In general, at node i the mth shape function is Function from “local basis” providing enrichment (e.g., higher order consistency, asymptotic solutions etc) PU function providing local support e.g., to obtain linear accuracy of the approximation, use 2 shape functions per node

  9. PU Method : Choice of local approx spaces • Shephard function-based PU • Polynomial PU • Generalized FEM (GFEM) • Extended FEM (XFEM)

  10. PU Method : Link to Taylor series In general, any function u(x) may be written as Now expand u(x) about xi If you truncate this series, then it becomes an approximation to u(x), i.e. where are coefficients replacing

  11. Shape functions e.g. Step1: Generate a set of functions that guarantee rigid body modes Partition of Unity (PU) Method in 3D xI WI Radius rI I Shepard PU function W Step 2: Define local spaces :Open ball of radius rI and center xI. Step 3: Global approximation space J. M. Melenk and Babuska, The Partition of Unity Finite Element Method: Basic Theory and Applications, Comp Methds in App Mech and Engg, 139, 289-314 (1996). C. A. Duarte and J. T. Oden, H-p Clouds--an h-p Meshless Method, Num. Meth. Partial Differential Equations, 12, 673-705 (1996).

  12. 0.3 -0.3 -1 -1 y x 1 1 In a nutshell Shape function at node I corresponding to mth d.o.f Polynomial or other function Shepard PU function

  13. Partition of Unity (PU) Method Properties • Reproducing property: Any function pm(x) included in the local basis of ALL the nodes can be exactly reproduced. • Continuity : If and then

  14. W1(x) W2(x) W0(x) Wn(x) x2 x0 x1 xn x Partition of Unity (PU) Method Kronecker delta The PU shape functions do not usually satisfy the Kronecker delta property. Special construction: for any node ‘j’ then

  15. x x1=0 x3=1 x2=?? PU and enrichment Question: How do you incorporate a strain singularity of in a single quadratic (isoparametric) finite element in 1D? Recap: Isoparametric coordinates Isoparametric map s 0 1 -1 Displacement approximation Strain approximation where

  16. PU and enrichment Question: How can you tailor the FEM approximation space to represent the function sin(x)? Not so simple! Use the PU concept Define 4 shape functions 2 1 FEM Approximation u1 and u2 are the nodal displacements at nodes 1 and 2

  17. PU and enrichment Question: How can you enrich a Shephard PU-based approximation locally to represent a strain singularity PU Approximation x1=0 x2 x3 x3

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