# Andy R Terrel November 8-9, 2006 FEniCS 06 Delft University of Technology

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1 Andy R Terrel Department of Computer Science University of Chicago November 8-9, 2006 FEniCS 06 Delft University of Technology

2 Outline 1 2 3

3 Motivation Success of the Finite Element Method(FEM) has led to a proliferation of FEM simulation software. The FEniCS Project, Sundance, DEAL Others: FreeFEM, FEMLab,... No single package meets everyone s needs... yet. Sundance handles optimization well, but is limited in kinds of elements, FEniCS gives a good smaller bundle that effeciently generates code, DEAL handles a more elements and has a larger user community, Both Sundance and FEniCS use easily readable code for user input.

4 There seem to still be disjunctions between the math and the software. Domains and meshes are not always identical. Small variations of methods are hard or not possible.... other issues... Some goals for this talk Point out some rough spots in current software through my experiences. Get feedback on validity and feasibility of ideas. Challenge the next generation of FEM software to be more mathematically rigorous Rigorous in code correctness. Rigorous in correct mathematical abstractions from problems.

5 Analysis v. Simulation Mathematically a finite element method is simply: A reference element, K A space of shape functions, P A basis, N But it seems there is something missing: Links between elements, How the elements affect the solvers.

6 Alternate Methods It seems there is always someone who wants to do something different. How much control of you software to give the user, to play with new methods? Example: Mixed methods, do we just put the formulation in the software or give the user the matrices.

7 y Example: Stokes Equations u + p = f u = 0, u = [ sin(πx) cos(πy) cos(πx) sin(πy) Using Taylor-Hood elements with mixed formulation, ] Number of Iterations mesh P1/P2 P2/P3 4x x x x Stokes velocity x

8 How about optimization problems? Use Automatic Differentiation tools on produced code - expensive on user side Create a symbolics engine that can give derivatives - expensive on developer side Example Problem in Microfluidic Devices To optimize flow, change channel geometry Most effective methods, use level set methods

9 What meshes can we handle? None of these software packages are giving us great tools for multigrid adaptivity. - courtesy Peter Brune.

10 Why is Mathematical Software Hard? The design space for mathematical software is multi-dimensional and not orthogonal. Mesh: uniform meshes, general geometry, adaptive meshes, unstructured meshes Function Space: linears, menu of options, arbitrary order, FIAT, exterior calculus Equation Description: menu, language, derived forms, error estimators Solver algorithms: menu, language Parallel Computing Support Boundary Conditions and embedded geometries.

11 Future Good mathematical abstractions have gotten us this far, where else can we go? How do these issues play well with software design principles? Is there a single solution to automated mathematical modelling? Questions or Comments?

12 What about code complexity? One bad estimate is lines of code: Dolfin + FFC 50K lines Sundance 100K lines DEAL 400K lines Some Organization Charts

13 More Detailed Dependencies An example of Dependencies for Sundance (not especially different from others)

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