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，且与样条逼近的多项式阶数无关。为离散测试函数空间，该方法采用近似对偶基，其函数具有光滑性、局部支撑性，并与B样条试验空间满足近似双正交性。由此生成的质量矩阵“接近”单位矩阵。具体而言，该质量矩阵的集中版本在用于Galerkin投影时能保留所有相关多项式。因此，通过行求和集中法对该质量矩阵进行集中处理，不会降低显式动力学计算中的空间精度。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，通过迭代算法建立了精确对偶基与近似对偶基函数之间的关联，该算法可使近似对偶基逐步趋近于精确双正交性。通过离散梁、板和壳模型的收敛性研究及谱分析，验证了我们高阶精确质量集中方法的性能。