Finite Elements in multidimensions Sauro Succi Finite Elements

Finite Elements in multi-dimensions Sauro Succi

Finite Elements: full power! The power of FEM remains somehow silent in d=1, much more visible in d=2 and d=3, where geometrical complexity is often the key factor

Capturing any shape… The local order of the polynomial matches the number of nodes

High-Order FEM

2 D FEM: structural mechanics Local deformation of a solid body obeys forced wave equation (small amplitudes): FEM “pyramid functions” (2 d hat): Leads to semidiscrete ODE’s:

2 D FEM: triangles and pyramids Six nodes=vertices for each hexagonal element, i=1, 6. Each pyramid overlaps with six neighbor pyramids. Three parameters on each sub-element (Triangle): Linear

Computing on triangles Global (x, y) to local controvariant-coordinates

Contro-variant coordinates Global (x, y) to Local (r, s) con-coordinates

Triangles: explicit transform Global (x, y) to Local (r, s) coordinates Generally not needed, but nonetheless:

Integrals on Triangles Global (x, y) to Local (r, s) con-coordinates Unit Triangle U

Mass matrix elements And so on for a total of 9={0, 1, 2}x{0, 1, 2}

Matrix elements: quadrature How about generic integrals of the form? Quadrature Formulas on Triangles: Order 1: exact for {1, r, s} Order 2: exact for {1, r, s, r*r, r*s, s*s}

Stiffness matrix elements Each triangle sharing i and j as vertices contributes to the matrix element (i, j): Now the area is not sufficient, local metric factors {a, b, c, d} enter explictly

Matrix assembly Each finite-element centered on the node j, connects to Z(j) neighbors. Each sub-element (triangle) contains 3 nodes: {x 1, y 1; x 2, y 2; x 3, y 3} and a pointer to the global address: p(e, 1)=i 1, p(e, 2)=i 2, p(e, 3)=i 3. Sub-elements are the fundamental geometric units! For all e=1, Nel The list of node coordinates, elements, connectivicty and metric factors must be prepared before hand: MESHING. This is the most crucial step (Pre-Processing), since the quality of the mesh is key to a successful simulation!

Element-wise assembly Local to Global matrix: The local to global mapping must be designed so as to Minimise the Logical distance {i-j} over the entire mesh. This is key for direct methods, but always beneficial. Optimal numbering is a NP-complete problem! (Travel-sales man-like)

Matrix assembly

Linear Algebra Direct Methods: Minimize bandwidth Optimal Numbering (NP complete) Iterative Methods: Sparse matrix algebra: y=A*x

Optimal numbering 13 11 18 20 12 19 9 10 1 2 3 13 14 16 12 15 8 17 4 15 14 7 5 6 2 1 3 4 7 11 10 9 8 5 Matrix bandwidth: Computer storage is one-dimensional! Hence a 3 d structure must be projected down to a linear array. As a result, the logical distance (computer memory) differs drastically from the geometric distance in physical space: whence the optimal numbering problem. 6

Geo-Topological Data Metric File Nodes 1 2 Node Coordinates x 1, y 1, z 1 x 2, y 2, z 2 … N …. . x. N, y. N, z. N Node Id nid(1)={0, 1, 2} 11 nid(N) Topology File Element 1 2 Three nodes indices i(1), j(1) k(1) i(2), j(2), k(2) … Nel …. . i(Nel) j(Nel), k(Nel) 20 10 9

Sparse FEM matrices This structure refelects the geometrical connectivity (# of non-zero elements per row) and the Numbering policy (Bandwidth)

3 D FEM

3 D FEM: T 3=Tetrahedra

3 D FEM: Tetrahedra The logical procedure stays the same, but all stages of the calculation are more laborious, in all respects: 4 x 4=16 local matrix elements {i, j, k, l} More metric factors within the T 3=Tetrahedron Larger matrices O(N^3) and bandwidths B=O(N^2) Solid but heavy-duty!

FEM for Navier-Stokes

FEM for NSE: Nonlinearity p is the OUTER iteration, typically the starting point is the value at the previous time-step Not to be confused with the INNER iteration, the one to solve Ax=b by iterative methods (see previous lecture)

FEM operators + Strong math back-up + Very systematic + Fluid/Solid coupling - Expensive (matrix algebra)

Summary FEM + Geometrical flexibility, adaptivity Powerful math backup (weak convergence) Systematic programming. Once the Elements are chosen, all follows automatically no decisions required. FEM Matrix algebra anyway (lumping helps) Heavy duty Bottomline: Mainstream for solid mech, fluid-solid but less for fluids alone.

Open Source and Commercial FEM for Fluids/Solids Open. FOAM (Field Operation and Manipulation) Listed among FEM but it’s FVM? !? COMSOL Multiphysics: (see Wikipedia) Set of flows, Core Numerical methods: Numerical Solver: It is candid about FVM being more popular (simpler) FEM is unsurpassed for structural mechanics (no advection…)

COMSOL

Multi-elements, adapted to The operators (Multiphysics)

Assignements 1. Solve Poisson in a d=2 rectangle using cartesian hat functions (see next slides) 2. Same as 1. using triangles 3. Same as 1 and/or 2. for 2 D Schroedinger

Cartesian 2 d FEM The FEM is composed of 4 sub-elements (triangle). Each sub-element generates 4 x 4=16 local matrix elements. The global matrix contains 9 non-zero elements per row

Cartesian 2 d FEM Since the element is a direct product of x*y 1 d elements, all matrix elements are separable.

Example: 2 d mass matrix Same goes for all other matrices

End of the Lecture