Fast and accurate nonlinear hyper‐elastic deformation with a posteriori numerical verification of the convergence of solution: Application to the simulation of liver deformation,International Journal for Numerical Methods in Biomedical Engineering

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Fast and accurate nonlinear hyper‐elastic deformation with a posteriori numerical verification of the convergence of solution: Application to the simulation of liver deformation
International Journal for Numerical Methods in Biomedical Engineering ( IF 2.2 ) Pub Date : 2021-02-19 , DOI: 10.1002/cnm.3444
Farah Saidi ₁ , Abed Malti ₂

Affiliation

In this paper, we propose a new method to reduce the computational complexity of calculating the tangential stiffness matrix in a nonlinear finite element formulation. Our approach consists in partially updating the tangential stiffness matrix during a classic Newton–Raphson iterative process. The complexity of such an update process has the order of the number of mesh vertices to the power of two. With our approach, this complexity is reduced to the power of two of only the number of updated vertices. We numerically study the convergence of the solution with our modified algorithm. We describe the deformation through a strain energy density function which is defined with respect to the Lagrangian strain. We derive the conditions of convergence for a given tangential stiffness matrix and a given set of updated vertices. We use nonlinear geometric deformation and the nonlinear Mooney‐Rivilin model with both tetrahedron and hexahedron element meshing. We provide extensive results using a cube with small and large number of elements. We provide results on nonlinearly deformed liver with multiple deformation ranges of updated vertices. We compare the proposed method to state‐of‐the‐art work and we prove its efficiency at three levels: accuracy, speed of convergence and small radius of convergence.

中文翻译：

具有解收敛性的后验数值验证的快速准确的非线性超弹性变形：在肝脏变形模拟中的应用

在本文中，我们提出了一种新方法来降低非线性有限元公式中计算切向刚度矩阵的计算复杂度。我们的方法包括在经典的 Newton-Raphson 迭代过程中部分更新切向刚度矩阵。这种更新过程的复杂性是网格顶点数量的 2 次方。通过我们的方法，这种复杂性降低到更新顶点数量的 2 次方。我们用我们的修改算法数值研究了解决方案的收敛性。我们通过相对于拉格朗日应变定义的应变能密度函数来描述变形。我们推导出给定切向刚度矩阵和一组给定更新顶点的收敛条件。我们使用具有四面体和六面体单元网格的非线性几何变形和非线性 Mooney-Rivilin 模型。我们使用具有少量和大量元素的立方体提供了广泛的结果。我们提供了具有多个更新顶点变形范围的非线性变形肝脏的结果。我们将所提出的方法与最先进的工作进行了比较，并在三个层面证明了其效率：准确性、收敛速度和小收敛半径。

更新日期：2021-02-19

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