Title: Preserving the positivity of the deformation gradient determinant in intergrid interpolation by combining RBFs and SVD: application to cardiac electromechanics Show Chinese title

Title: 通过结合径向基函数和奇异值分解在网格间插值中保持变形梯度行列式的正性：在心脏电机械学中的应用

Abstract: The accurate robust and efficient transfer of the deformation gradient tensor between meshes of different resolution is crucial in cardiac electromechanics simulations. We present a novel method that combines rescaled localized Radial Basis Function (RBF) interpolation with Singular Value Decomposition (SVD) to preserve the positivity of the determinant of the deformation gradient tensor. The method involves decomposing the evaluations of the tensor at the quadrature nodes of the source mesh into rotation matrices and diagonal matrices of singular values; computing the RBF interpolation of the quaternion representation of rotation matrices and the singular value logarithms; reassembling the deformation gradient tensors at quadrature nodes of the destination mesh, to be used in the assembly of the electrophysiology model equations. The proposed method overcomes limitations of existing interpolation methods, including nested intergrid interpolation and RBF interpolation of the displacement field, that may lead to the loss of physical meaningfulness of the mathematical formulation and then to solver failures at the algebraic level, due to negative determinant values. The proposed method enables the transfer of solution variables between finite element spaces of different degrees and shapes and without stringent conformity requirements between different meshes, enhancing the flexibility and accuracy of electromechanical simulations. Numerical results confirm that the proposed method enables the transfer of the deformation gradient tensor, allowing to successfully run simulations in cases where existing methods fail. This work provides an efficient and robust method for the intergrid transfer of the deformation gradient tensor, enabling independent tailoring of mesh discretizations to the particular characteristics of the physical components concurring to the of the multiphysics model. Show Chinese abstract

Abstract: 在心脏电机械模拟中，不同分辨率网格之间变形梯度张量的准确、鲁棒和高效传递至关重要。我们提出了一种新方法，将缩放局部径向基函数（RBF）插值与奇异值分解（SVD）相结合，以保持变形梯度张量行列式的正性。该方法涉及将源网格的高斯节点处张量的评估分解为旋转矩阵和奇异值对角矩阵；计算旋转矩阵的四元数表示和奇异值对数的RBF插值；在目标网格的高斯节点处重新组装变形梯度张量，用于电生理学模型方程的组装。所提出的方法克服了现有插值方法的局限性，包括嵌套的网格间插值和位移场的RBF插值，这些方法可能导致数学公式的物理意义丧失，并由于负行列式值导致代数层面的求解器失败。所提出的方法实现了不同次数和形状的有限元空间之间的解变量传递，且不同网格之间无需严格的符合性要求，增强了电机械模拟的灵活性和准确性。数值结果证实，所提出的方法实现了变形梯度张量的传递，使得在现有方法失败的情况下能够成功运行模拟。这项工作提供了一种高效且鲁棒的变形梯度张量网格间传递方法，使得可以独立地根据参与多物理场模型的物理组件的特定特性来定制网格离散化。