G-Adaptivity: optimised graph-based mesh relocation for finite element methods
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arXiv
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| Autores principales: | , , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909654874849280 |
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| author | Rowbottom, James Maierhofer, Georg Deveney, Teo Mueller, Eike Paganini, Alberto Schratz, Katharina Liò, Pietro Schönlieb, Carola-Bibiane Budd, Chris |
| author_facet | Rowbottom, James Maierhofer, Georg Deveney, Teo Mueller, Eike Paganini, Alberto Schratz, Katharina Liò, Pietro Schönlieb, Carola-Bibiane Budd, Chris |
| contents | We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04516 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | G-Adaptivity: optimised graph-based mesh relocation for finite element methods Rowbottom, James Maierhofer, Georg Deveney, Teo Mueller, Eike Paganini, Alberto Schratz, Katharina Liò, Pietro Schönlieb, Carola-Bibiane Budd, Chris Machine Learning Numerical Analysis We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work. |
| title | G-Adaptivity: optimised graph-based mesh relocation for finite element methods |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2407.04516 |