Mike Hillman

Karagozian and Case, Inc.

Because most approximation functions employed in meshfree methods are rational functions with overlapping supports, sufficiently accurate domain integration becomes costly, whereas insufficient accuracy in the domain integration leads to suboptimal convergence. In this paper, we show that it is possible to achieve optimal convergence by enforcing v...

State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree form...

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) is introduced for solving the governing equations of generalized thermo-mechanical theories. Part I investigates quadrature in the weak form using coupled and uncou-pled classical thermoelasticity as model problems. It is first shown that nod...

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) approach for solving the governing equations of generalized thermomechanical theories is developed. Part I investigated quadrature in the weak form using classical thermoe-lasticity as a model problem, and a stabilized and corrected nodal int...

Generating quality body-fitting meshes for complex composite microstructures is a non-trivial task. In particular , micro-CT images of composites can contain numerous irregularly-shaped inclusions. Among the methods available, immersed boundary methods that discretize bodies independently provide potential for tackling these types of problems since...

We extend the recently proposed framework using reduced quadrature in the Finite Element and Isogeometric methods for solid mechanics to the nonlinear realm. The proposed approach makes use of the governing equations in the updated Lagrangian formulation in combination with the rate form of the constitutive laws. The key ingredient in the framework...

Non-symmetric matrices may arise in the discretization of self-adjoint problems when a Petrov–Galerkin, collocation, or finite-volume method is employed. While these methods have been widely applied, in this paper it is shown that the use of these non-symmetric matrices is incompatable with the conservation of energy in elastodynamics. First, the c...

Highlights • Analysis of energy-momentum consistency on continuous level and discrete level • Temporal stability analysis in solving elastodynamics by non-symmetric type methods • Unconditionally unstable solutions for Petrov-Galerkin methods shown • A time integration is proposed to exactly conserve energy in non-symmetric systems Abstract Non-sym...

We present a novel formulation for the immersed coupling of isogeometric analysis and peridynamics for the simulation of fluid–structure interaction (FSI). We focus on air-blast FSI and address the computational challenges of immersed FSI methods in the simulation of fracture and fragmentation by developing a weakly volume-coupled FSI formulation b...

Damage modes are drastically different for concrete under complex stress states. This study investigates damage in high-strength concrete under triaxial loading with confinement pressures up to 200 MPa, while also considering effects from changes in specimen length-to-diameter ratio. Damage was observed and segmented using X-ray microtomography. Hy...

We present a novel formulation for the immersed coupling of Isogeometric Analysis (IGA) and Peridynamics (PD) for the simulation of fluid-structure interaction (FSI). We focus on air-blast FSI and address the computational challenges of immersed FSI methods in the simulation of fracture and fragmentation by developing a weakly volume-coupled FSI fo...

Generating quality body-fitting meshes for complex composite microstructures is a non-trivial task. In particular, micro-CT images of composites can contain numerous irregularly-shaped inclusions. Among the methods available, immersed boundary methods that discretize bodies independently provide potential for tackling these types of problems since...

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) is introduced for solving the governing equations of generalized thermomechanical theories. Part I investigates quadrature in the weak form using coupled and uncoupled classical thermoelasticity as model problems. It is first shown that nodal...

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) approach for solving the governing equations of generalized thermomechanical theories is developed. Part I investigated quadrature in the weak form using classical thermoelasticity as a model problem, and a stabilized and corrected nodal inte...

The explosive welding process is an extreme-deformation problem that involves shock waves, large plastic deformation, and fragmentation around the collision point, which are extremely challenging features to model for the traditional mesh-based methods. In this work, a particle-based Godunov shock algorithm under a semi-Lagrangian reproducing kerne...

Polymer-ceramic composites are widely used in biomedical applications. This paper presents the results of an experimental investigation on the crack extension inside epoxy-alumina. Specimens with 5 vol.%, 10 vol.%, …, 25 vol.% fillers fractions were fabricated. Three-point bending on single-edge notched bend specimens were performed using conventio...

Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique to modify the a...

Highlights • Two weak forms are introduced that are consistent with meshfree approximations • Higher order optimal h-refinement previously unavailable • p-refinement previously unavailable • New ability to increase accuracy called a-refinement Abstract Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial d...

Meshfree methods such as the reproducing kernel particle method (RKPM) are well suited for modeling materials and solids undergoing fracture and damage processes, and nodal integration is a natural choice for modeling this class of problems. However, nodal integration suffers from spatial instability, and the excessive material deformation and dama...

State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree form...

We present an open-source software RKPM2D for solving PDEs under the reproducing kernel particle method (RKPM)-based meshfree computational framework. Compared to conventional mesh-based methods, RKPM provides many attractive features, such as arbitrary order of continuity and discontinuity, relaxed tie between the quality of the discretization and...

The theory and meshfree implementation of peridynamics has been proposed to model problems involving transient strong discontinuities such as dynamic fracture and fragment-impact problems. For effective application of numerical methods to these events, essential shock physics and Gibbs instability should be addressed. The Godunov scheme for shock t...

The viscoelastic properties of ASR gels have a major impact on the gels' restrained swelling pressure and the resulting damage to concrete. This study employed a design of experiments approach to synthesize and test eighteen cylindrical ASR gel specimens of different compositions, expressed in terms of their Ca/Si, Na/Si and K/Si molar ratios. The...

For extremely large deformation problems in solid mechanics, the Lagrangian finite element approach is often ineffective. Galerkin meshfree methods developed extensively over the past twenty years and are sufficiently mature to handle simulations involving large deformations with relative ease. Many state-of-the-art advances in these methods have b...

In the process of Data-Driven modeling of the material microstructures, a reduced-order representation is preferred, as it is more manageable from the numerical standpoint. Its main drawback though is the loss of the connection between the parameters of the reduced-order model and the physical properties of the microstructure. We propose a methodol...

The theory and meshfree implementation of peridynamics has been proposed to model problems involving transient strong discontinuities such as dynamic fracture and fragment-impact problems. For effective application of numerical methods to these events, essential shock physics and Gibbs instability should be addressed. The Godunov scheme for shock t...

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,...

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...

In this two-part paper we begin the development of a new class of methods for modeling fluid–structure interaction (FSI) phenomena for air blast. We aim to develop accurate, robust, and practical computational methodology, which is capable of modeling the dynamics of air blast coupled with the structure response, where the latter involves large, in...

In 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 addressing the major shortcomings that were present in these methods at the early stages of their development. For instance, essential boundary conditions are almo...

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 different...

Convective transport terms in Eulerian conservation laws lead to numerical instability in the solution of Bubnov-Galerkin methods for these non-self-adjoint PDEs. Stabilized Petrov-Galerkin methods overcome this difficulty, however gradient terms are required to construct the test functions, which are typically expensive for meshfree methods. In th...

A meshfree formulation under the reproducing kernel particle method (RKPM) was introduced for modeling the penetration and perforation of brittle geomaterials such as concrete. RKPM provides a robust framework to effectively model the projectile-target interaction and the material failure and fragmentation behaviors that are critical for this class...

The reproducing kernel particle method (RKPM) is a meshfree method for computational solid mechanics that can be tailored for an arbitrary order of completeness and smoothness. The primary advantage of RKPM relative to standard finite-element (FE) approaches is its capacity to model large deformations, material damage, and fracture. Additionally, t...

doi:10.1002/nme.5183 Accepted manuscript online: 5 December 2015 Manuscript Accepted: 27 November 2015 Manuscript Revised: 11 November 2015 Manuscript Received: 19 August 2015 Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stabili...

This document is a pre-print version of a contribution to Modeling and Simulation in Science, Engineering and Technology Book Series devoted to AFSI 2014 - a birthday celebration conference for Tayfun Tezduyar. Submitted Sep. 24, 2015. Accepted Nov. 19, 2015. Meshfree approximations are ideal for the gradient-type stabilized Petrov- Galerkin metho...

Numerical modeling of reservoirs with low permeability or under undrained conditions often suffers from spurious fluid pressure oscillations due to the improper construction of approximation spaces. To address this issue, a fully coupled, stabilized meshfree formulation is developed based on a fluid pressure projection method, in which an additiona...

Spline-type approximations for solving partial differential equations are the basis of isogeometric analysis. While the common approach of using integration cells defined by single knot spans using standard (e.g., Gaussian) quadrature rules is sufficient for accuracy, more efficient domain integration is still in high demand. The recently introduce...

Galerkin meshfree methods can suffer from instability and suboptimal convergence if the issue of quadrature is not properly addressed. The instability due to quadrature is further magnified in high strain rate events when nodal integration is used. In this paper, several stable and convergent nodal integration methods are presented and applied to t...

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 met...