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Study of solver strategies to lead with�boundary conditions in multi-physics�problems

BACHELOR’S FINAL THESIS

AUTHOR: PAU GIL CORRETGER

DIRECTOR: ÀLEX FERRER FERRÉ

CODIRECTOR: TON CREUS COSTA

BACHELOR’S DEGREE IN AEROSPACE TECHNOLOGIES ENGINEERING

17/07/2023

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Index

  • Introduction
  • Types of boundary conditions
  • Reduced formulation
  • Monolithic formulation
  • Proposed contribution
  • Result comparison
  • Final contributions
  • Conclusions and future work
  • Bibliography

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Introduction

  • Aim: To develop a general methodology to solve multi-physics problems.
  • Most problems can be tackled from a Boundary Conditions (BC) point of view.
  • Methodology to be applied in topology optimization algorithms.
  • Multiple examples in modern engineering problems.

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Heat transfer problems

Structural analysis

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Types of boundary conditions

  • Dirichlet
    • Variables with already known values in the affected domain.
    • Values may be null or non null.
  • Periodic
    • Studied cell surrounded by identical neighbors.
    • Boundary variables forced to have the same values.
    • Useful in microstructures.
  • Mixed problems
    • Combination of both types

Dirichlet BC

Nature of periodic BC

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Reduced formulation

  • Dirichlet BC:
    • Weak form:

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    • Direct solution does not return the reaction vector.
  • Periodic BC:

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    • Direct solution does not return the homogenized stress vector.

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Monolithic formulation

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Discretized form

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Monolithic formulation

  • Approach for microstructures
    • Fluctuations point of view:

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    • Global displacements point of view:

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Displacement composition

Homogenized stress computation

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Proposed contribution

  • Code modification on Swan’s GitHub repository.
  • Possibility to solve multiple problems by specifying the solver characteristics in the input file.
  • Differentiation between reduced/monolithic formulations and periodic/global approaches.
  • Computation of constitutive tensors from Lagrange multipliers.

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Result comparison

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  • Cantilever: Non null Dirichlet BC

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Result comparison

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  • Total displacements

Periodic conditions

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Result comparison

  •  

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  • Total displacements

Periodic conditions

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Result comparison

  • Lagrange multiplier distribution

 

 

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Result comparison

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  • Total displacements: Horizontal axis

Minimum conditions

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Final contributions

Before

After

BC class only working with reduced formulations

BC child classes that make additional assemblies (monolithic)

Solver for macrostructures only working for null Dirichlet values

Solver for macrostructures solves 2D problems for every type of Dirichlet constraint

Additional algorithms required to compute constitutive tensors

Constitutive tensors computed by simple Lagrange multipliers summations

Available test platform for reduced solvers

Created test platform for every type of solver including the necessary input data

Initial implemented architecture difficult to follow

More user-friendly architecture while applying clean code practices

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Conclusions and future work

Conclusions

  • Monolithic solvers are a real alternative to Gauss elimination methods.
  • Lagrange multipliers give reliable constitutive tensors and reaction vectors.
  • Object Oriented Programming (OOP) implemented correctly to build a new architecture.

Future work

  • Extension to more types of BC: Robin…
  • Extension to 3D cases.
  • Implementation and testing of new alternatives: Schur complement.

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Bibliography

  1. CREUS COSTA, Ton. Study of numerical methods for the design of large lightweight structures [online]. 2022. [visited on 2023-02-21]. Available from: https://upcommons.upc.edu/ handle/2117/372281. Bachelor thesis. Universitat Polit`ecnica de Catalunya. Accepted: 2022- 09-06T07:08:52Z. 2.
  2. SwanLab/Swan [online]. SwanLab, 2023 [visited on 2023-03-01]. Available from: https:// github.com/SwanLab/Swan. original-date: 2017-09-20T15:04:37Z.
  3. Abaqus Unified FEA - SIMULIA™ de Dassault Systèmes® [online]. [visited on 2023-03-08]. Available from: https://www.3ds.com/es/productos-y-servicios/simulia/productos/ abaqus/.
  4. firedrakeproject/firedrake [online]. Firedrake, 2023 [visited on 2023-03-02]. Available from: https: //github.com/firedrakeproject/firedrake. original-date: 2013-10-07T13:07:44Z.
  5. BOURAQADI, Noury; MASON, Dave. Test-driven development for generated portable Javascript apps. Science of Computer Programming [online]. 2018, vol. 161, pp. 2–17 [visited on 2023-05-17]. issn 0167-6423. Available from doi: 10.1016/j.scico.2018.02.003.
  6. Fundamental interface for grouping tests to run - MATLAB - MathWorks España [online]. [visited on 2023-05-17]. Available from: https://es.mathworks.com/help/matlab/ref/ matlab.unittest.testsuite-class.html.
  7. GORBUNOV, S. A.; VOLKOV, A. E.; VORONKOV, R. A. Periodic boundary conditions effects on atomic dynamics analysis. Computer Physics Communications [online]. 2022, vol. 279, p. 108454 [visited on 2023-03-23]. issn 0010-4655. Available from doi: 10.1016/j.cpc.2022. 108454.

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