We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is ``close'' to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

翻译：我们提出了一种基于等几何Petrov-Galerkin方法的质量集中方案，该方法在显式动力学计算中能够保持高阶空间精度，且不受样条逼近多项式阶次的影响。该方案采用近似对偶基离散试探函数空间，其函数具有光滑性、局部支撑性，并与B样条试空间满足近似双正交关系。由此生成的质量矩阵近乎单位矩阵。具体而言，当该质量矩阵的集中版本用于Galerkin投影时，能够保留所有相关多项式。因此，通过行求和集中法压缩质量矩阵不会损害显式动力学计算的空间精度。我们解决了狄利克雷边界条件的施加问题，以及几何映射下近似双正交性的保持问题。此外，我们通过迭代算法建立了精确对偶基函数与近似对偶基函数之间的联系，该算法可使近似双正交性逐步趋近精确双正交性。通过离散梁、板和壳模型的收敛性研究与频谱分析，验证了所提出的高阶精确质量集中方法的有效性。