The a posteriori finite element method (APFEM), a framework for efficient parametric study and Bayesian inferences
The a posteriori finite element method (APFEM), a framework for efficient parametric study and Bayesian inferences
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DOI：
10.1016/j.cma.2023.115996
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发表时间：
2023-07
影响因子：
7.2
通讯作者：
Y. Ammouche;A. Jérusalem
中科院分区：
文献类型：
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作者：
Y. Ammouche;A. Jérusalem
随机方法最近在计算力学中受到越来越多的关注，因为它们能够在预测中考虑材料参数和几何特征的随机性。其中，Galerkin随机有限元法（GSFEM）被证明是特别有效的，能够提供准确的输出统计，虽然在侵入性编码和额外的理论代数努力的成本。在该方法中，随机参数的分布被用作求解器的输入，求解器进而在一次模拟中输出节点位移分布。在这里，我们提出了一个扩展的GSFEM-称为后验有限元法或APFEM-在默认情况下采取均匀分布，以允许参数化的研究作为后处理步骤的输入感兴趣的模拟后。这样做，APFEM只需要参数空间的顶点的知识。特别是，APFEM的一个关键优势是它在贝叶斯推理的背景下使用，其中贝叶斯设置所需的随机评估（通常通过蒙特卡罗完成）可以精确地完成，而不需要进一步的模拟。最后，我们证明了APFEM的潜力，通过求解前向模型的参数边界条件的背景下（i）超材料的设计和（ii）音叉分叉的细长结构的屈曲;并证明其使用贝叶斯推理的灵活性（iii）推断摩擦系数的半平面接触力学问题和（iv）在癌症手术计划的背景下推断大脑区域的硬度。
Stochastic methods have recently been the subject of increased attention in Computational Mechanics for their ability to account for the stochasticity of both material parameters and geometrical features in their predictions. Among them, the Galerkin Stochastic Finite Element Method (GSFEM) was shown to be particularly efficient and able to provide accurate output statistics, although at the cost of intrusive coding and additional theoretical algebraic efforts. In this method, distributions of the stochastic parameters are used as inputs for the solver, which in turn outputs nodal displacement distributions in one simulation. Here, we propose an extension of the GSFEM—termed theA posterioriFinite Element Method or APFEM—where uniform distributions are taken by default to allow for parametric studies of the inputs of interest as a postprocessing step after the simulation. Doing so, APFEM only requires the knowledge of the vertices of the parameter space. In particular, one key advantage of APFEM is its use in the context of Bayesian inferences, where the random evaluations required by the Bayesian setting (usually done through Monte Carlo) can be done exactly without the need for further simulations. Finally, we demonstrate the potential of APFEM by solving forward models with parametric boundary conditions in the context of (i) metamaterial design and (ii) pitchfork bifurcation of the buckling of a slender structure; and demonstrate the flexibility of its use for Bayesian inference by (iii) inferring friction coefficient of a half plane in a contact mechanics problem and (iv) inferring the stiffness of a brain region in the context of cancer surgical planning.