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Finite elements for Mat\'ern-type random fields: Uncertainty in computational mechanics and design optimization
Computational Engineering, Finance, and Science · Submitted 6 Mar 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.
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The abstract, sentence by sentence
This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on applications in computational mechanics and design optimization. We leverage Mat\'ern-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models and the topology of optimized designs. We describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications. The flexibility and efficiency of SPDE-based GRF generation empower us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model geometric uncertainties of reconstructed submanifolds, such as the surfaces of cerebral aneurysms. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on applications in computational mechanics and design optimization. | contribution 0.88 | 0.56 | 0.15 | asserted 0.99 |
| s1 | We leverage Mat\'ern-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. | method 0.96 | 0.83 | 0.06 | asserted 1.00 |
| s2 | Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models and the topology of optimized designs. | contribution 0.93 | 0.57 | 0.16 | asserted 0.85 |
| s3 | We describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. | method 0.98 | 0.83 | 0.05 | asserted 1.00 |
| s4 | The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. | method 0.84 | 0.55 | 0.04 | asserted 1.00 |
| s5 | Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. | method 0.39 | 0.79 | 0.06 | asserted 1.00 |
| s6 | We showcase its versatility through biomechanics and topology optimization applications. | contribution 0.78 | 0.86 | 0.07 | asserted 1.00 |
| s7 | The flexibility and efficiency of SPDE-based GRF generation empower us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate topologies beyond the reach of conventional techniques. | interpretation 0.36 | 0.85 | 0.35 | asserted 0.99 |
| s8 | Moreover, these capabilities allow us to model geometric uncertainties of reconstructed submanifolds, such as the surfaces of cerebral aneurysms. | contribution 0.86 | 0.73 | 0.06 | asserted 0.88 |
| s9 | In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements. | interpretation 0.82 | 0.49 | 0.40 | asserted 0.98 |
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The paper
Original abstract
This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on applications in computational mechanics and design optimization. We leverage Mat\'ern-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models and the topology of optimized designs. We describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications. The flexibility and efficiency of SPDE-based GRF generation empower us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model geometric uncertainties of reconstructed submanifolds, such as the surfaces of cerebral aneurysms. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.
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What this paper does with the arXiv papers it cites
What it cites · 8 sentences without a confident label
Uses the method or tool of 1 paper
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We implemented the SPDE method with MFEM~ [ref] , and the resulting MPI-parallel solver is part of MFEM's 4.6 release MFEM 4.6, https://mfem.org/download/mfem.org/download/, available in miniapps/spde.
High-performance finite elements with MFEM · Implementation and Scalability · 1.00
Cites as background 2 papers
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For a detailed discussion of the covariance of GRFs on the sphere, including the Matérn-type covariance, we refer to [ref] .
A diffusion-based spatio-temporal extension of Gaussian Mat\'ern fields · Optimization on embedded surfaces · 0.97
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Mathematical models for cerebral aneurysms and their treatment describe a complex interplay between fluid dynamics, biomechanics, and aneurysm geometry [ref] .
Numerical simulation of endovascular treatment options for cerebral aneurysms · Simulating random fields on complex domains · 0.87
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