Michael Hillman

Meshfree Methods: Progress Made after 20 Years

AbstractIn the past two decades, meshfree methods have emerged into a new class of computational methods with considerable success. In addition, a significant amount of progress has been made in ad...

Corrected Stabilized Non-conforming Nodal Integration in Meshfree Methods

A novel approach is presented to correct the error from numerical integration in Galerkin methods for meeting linear exactness. This approach is based on a Ritz projection of the integration error that allows a modified Galerkin discretization of the original weak form to be established in terms of assumed strains. The solution obtained by this method is the correction of the original Galerkin discretization obtained by the inaccurate numerical integration scheme. The proposed method is applied to elastic problems solved by the reproducing kernel particle method (RKPM) with first-order correction of numerical integration. In particular, stabilized non-conforming nodal integration (SNNI) is corrected using modified ansatz functions that fulfill the linear integration constraint and therefore conforming sub-domains are not needed for linear exactness. Illustrative numerical examples are also presented.

Performance Comparison of Nodally Integrated Galerkin Meshfree Methods and Nodally Collocated Strong Form Meshfree Methods

For a truly meshfree technique, Galerkin meshfree methods rely chiefly on nodal integration of the weak form. In the case of Strong Form Collocation meshfree methods, direct collocation at the nodes can be employed. In this paper, performance of these node-based Galerkin and collocation meshfree methods is compared in terms of accuracy, efficiency, and stability. Considering both accuracy and efficiency, the overall effectiveness in terms of CPU time versus error is also assessed. Based on the numerical experiments, nodally integrated Galerkin meshfree methods with smoothed gradients and variationally consistent integration yield the most effective solution technique, while direct collocation of the strong form at nodal locations has comparable effectiveness.

Numerical investigation of statistical variation of concrete damage properties between scales

Concrete is typically treated as a homogeneous material at the continuum scale. However, the randomness in micro-structures has profound influence on its mechanical behavior. In this work, the relationship of the statistical variation of macro-scale concrete properties and micro-scale statistical variations is investigated. Micro-structures from CT scans are used to quantify the stochastic properties of a high strength concrete at the micro-scale. Crack propagation is then simulated in representative micro-structures subjected to tensile and shear tractions, and damage evolution functions in the homogenized continuum are extracted using a Helmholtz free energy correlation. A generalized density evolution equation is employed to represent the statistical variations in the concrete micro-structures as well as in the associated damage evolution functions of the continuum. This study quantifies how the statistical variations in void size and distribution in the concrete microstructure affect the statistical variation of material parameters representing tensile and shear damage evolutions at the continuum scale. The simulation results show (1) the random variation decreases from micro-scale to macro-scale, and (2) the coefficient of variation in shear damage is larger than that in the tensile damage.

An Implicit Gradient Meshfree Formulation for Convection-Dominated Problems

Meshfree approximations are ideal for the gradient-type stabilized Petrov–Galerkin methods used for solving Eulerian conservation laws due to their ability to achieve arbitrary smoothness, however, the gradient terms are computationally demanding for meshfree methods. To address this issue, a stabilization technique that avoids high order differentiation of meshfree shape functions is introduced by employing implicit gradients under the reproducing kernel approximation framework. The modification to the standard approximation introduces virtually no additional computational cost, and its implementation is simple. The effectiveness of the proposed method is demonstrated in several benchmark problems.