Pai-Chen Guan

Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems

In this work, the weighted reproducing kernel collocation method (weighted RKCM) is introduced to solve the inverse Cauchy problems governed by both homogeneous and inhomogeneous second-order linear partial differential equations. As the inverse Cauchy problem is known for the incomplete boundary conditions, how to numerically obtain an accurate solution to the problem is a challenging task. We first show that the weighted RKCM for solving the inverse Cauchy problems considered is formulated in the least-squares sense. Then, we provide the corresponding error analysis to show how the errors in the domain and on the boundary can be balanced with proper weights. The numerical examples demonstrate that the weighted discrete systems improve the accuracy of solutions and exhibit optimal convergence rates in comparison with those obtained by the traditional direct collocation method. It is shown that neither implementation of regularization nor implementation of iteration is needed to reach the desired accuracy. Further, the locality of reproducing kernel approximation gets rid of the ill-conditioned system.

Solving Inverse Laplace Equation with Singularity by Weighted Reproducing Kernel Collocation Method

This work introduces the weighted collocation method with reproducing kernel approximation to solve the inverse Laplace equations. As the inverse problems in consideration are equipped with over-specified boundary conditions, the resulting equations yield an overdetermined system. Following our previous work, the weighted collocation method using a least-squares minimization has shown to solve the inverse Cauchy problems efficiently without using techniques such as iteration and regularization. In this work, we further consider solving the inverse problems of Laplace type and introduce the Shepard functions to deal with singularity. Numerical examples are provided to demonstrate the validity of the method.

Multiscale Semi-Lagrangian Reproducing Kernel Particle Method for Modeling Damage Evolution in Geomaterials

Damage processes in geomaterials typically involve moving strong and weak discontinuities, multiscale phenomena, excessive deformation, and multi-body contact that cannot be effectively modeled by a single-scale Lagrangian finite element formulation. In this work, we introduce a semi-Lagrangian Reproducing Kernel Particle Method (RKPM) which allows flexible adjustment of locality, continuity, polynomial reproducibility, and h- and p-adaptivity as the computational framework for modeling complex damage processes in geomaterials. Under this work, we consider damage in the continua as the homogenization of micro-cracks in the microstructures. Bridging between the cracked microstructure and the damaged continuum is facilitated by the equivalence of Helmholtz free energy between the two scales. As such, damage in the continua, represented by the degradation of continua, can be characterized from the Helmholtz free energy. Under this framework, a unified approach for numerical characterization of a class of damage evolution functions has been proposed. An implicit gradient operator is embedded in the reproduction kernel approximation as a regularization of ill-posedness in strain localization. Demonstration problems include numerical simulation of fragment-impact of concrete materials.

Semi-Lagrangian Galerkin Reproducing Kernel Formulation and Stability Analysis for Computational Penetration Mechanics

Stability analyses of Lagrangian and semi-Lagrangian reproducing particle methods using various domain integration methods are performed. The von Neumann stability analysis shows that both Lagrangian and semi-Lagrangian reproducing kernel discretizations of equation of motion are stable when they are integrated using stabilized conforming nodal integration in the weak forms. On the other hand, integrating the weak form of semi-Lagrangian equation of motion with a direct nodal integration yields an unstable discrete system which resembles the tensile instability in SPH. Stable time step estimation for Lagrangian reproducing kernel discretization shows enhanced stability when weak form is integrated by stabilized conforming nodal integration compared to that using direct nodal integration or 1-point Gauss integration. Penetration simulation is performed to demonstrate the applicability of the proposed method to large deformation and fragment impact problems.