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.

翻译：在心脏电力学模拟中，不同分辨率网格间变形梯度张量的精确、稳健且高效传递至关重要。本文提出一种结合重标定局部径向基函数插值与奇异值分解的新方法，以保持变形梯度张量行列式的正定性。该方法将源网格积分节点处的张量值分解为旋转矩阵与奇异值对角矩阵；对旋转矩阵的四元数表示和奇异值对数进行径向基函数插值；最终在目标网格积分节点处重构变形梯度张量，用于电生理模型方程组的组装。该方法克服了现有插值方法（包括嵌套网格间插值与位移场径向基函数插值）因负行列式值导致数学公式物理意义丧失及代数层面求解失败的局限性。该方法支持不同阶次和形状的有限元空间间变量传递，无需网格间严格共形条件，提升了电力学模拟的灵活性与精度。数值实验表明，该方法可实现变形梯度张量的可靠传递，成功运行现有方法失效的算例。本研究为变形梯度张量的网格间传递提供了高效稳健方法，使多物理场模型中各物理组件的网格离散化可根据其独特特性独立定制。