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Parallel two-scale finite element implementation of a system with varying microstructures

Omar Richardson, Omar Lakkis, Adrian Muntean, Chandrasekhar Venkataraman

TL;DR

This work proposes a two‐scale finite element method designed for heterogeneous microstructures using a conveniently constructed pullback operator to approximate the underlying system of partial differential equations with a parallel computational structure.

Abstract

We propose a two-scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying PDE system with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed-ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two-scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.

Parallel two-scale finite element implementation of a system with varying microstructures

TL;DR

This work proposes a two‐scale finite element method designed for heterogeneous microstructures using a conveniently constructed pullback operator to approximate the underlying system of partial differential equations with a parallel computational structure.

Abstract

We propose a two-scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying PDE system with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed-ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two-scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.

Paper Structure

This paper contains 17 sections, 1 theorem, 21 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A two-scale model with varying microstructures
  3. Model problem
  4. Assumptions on data
  5. Finite element implementation
  6. Approximation
  7. Implementation
  8. Structure of FEM implementation
  9. Domain mappings
  10. Parallelization in assembly
  11. Visualization
  12. Numerical experiments
  13. SymPy
  14. Convergence benchmarking
  15. Parallel benchmarking
  16. ...and 2 more sections

Key Result

Theorem 2.4

Under assumptions sse:assumptions.as:lips_omega--as:coerc system eq:macro-2xmicro-system admits a unique solution in the sense of def_solution which is also stable with respect to parameters.

Figures (10)

  • Figure 1: Schematic representation of the multiscale domain: at each macroscopic point $x\in\varOmega$ corresponds a microscopic domain $Y_x$ with mixed boundary conditions (pure Neumann or Robin).
  • Figure 2: Approximation error as a function of the degrees of freedom. The observed order of convergence is computed from the final step
  • Figure 3: Wall time (total duration of the simulation) as a function of the number of nodes for a fine-macro/coarse-micro system and a coarse-macro/fine-micro system.
  • Figure 4: Speed-up as a function of the number of nodes for a fine-macro/coarse-macro system and a coarse-macro/fine-micro system.
  • Figure 5: Nutrient concentration $u(x)$ in Case $A$, with the Dirichlet boundary (left) as the only source of nutrient.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Definition 2.2
  • Theorem 2.4: weak solvability
  • proof