High-rate deformation processes of metals such as explosive welding and cold spray additive manufacturing entail intense grain refinement. The multi-field variational formulation and the associated computational method capable of modeling the evolution of microstructures with sharp solution transition near the grain boundaries remain challenging in achieving high accuracy, stability, and computational efficiency. In this work, a new computational formulation for coupling Cosserat crystal plasticity and phase field is developed. The conventional approach by penalizing the kinematic incompatibility between lattice orientation and displacement-based elastic rotation leads to significant solution sensitivity to the penalty parameter, resulting in low accuracy and convergence rates. To address these issues, a duality-based formulation is developed under a multi-field variational framework. The associated Galerkin formulation incorporated with a weak inf-sup-based skew-symmetric stress projection is introduced to ensure coercivity for stability in the dual formulation. An additional least squares stabilization is introduced to suppress the spurious lattice rotation with a suitable parameter range derived analytically and validated numerically. It is shown that under this formulation, the equal order displacement-rotation-phase field approximations are stable, which allows efficient construction of approximation functions for all independent variables. The proposed formulation is shown to yield superior accuracy and convergence with marginal parameter sensitivity compared to the conventional penalty-based approach and successfully captures the dominant rotational recrystallization mechanisms that exist in the block dislocation structures and grain boundary migration.Modeling the sharp transition in the phase field near the grain boundaries associated with the lattice orientation often requires highly refined discretization for sufficient accuracy, which significantly increases the computational cost. While adaptive model refinement can be employed for enhanced effectiveness, it is cumbersome for the traditional mesh-based methods to perform adaptive model refinement. In this work, neural network-enhanced reproducing kernel particle method (NN-RKPM) is proposed, where the location, orientation, and the shape of the solution transition is automatically captured by the NN approximation by the minimization of total potential energy. The standard RK approximation is then utilized to approximate the smooth part of the solution to permit a much coarser discretization than the high-resolution discretization needed to capture sharp solution transition with the conventional methods. The proposed NN-RKPM is first verified by solving the standard damage evolution problems. The proposed computational framework is then applied to modeling grain refinement mechanisms, including the migration of grain boundaries at a triple junction, for validating the effectiveness of the proposed methods.

Biot's theory has been widely used to construct the poroelasticity models for describing the mechanical behavior of biological materials. This phenomenological framework, however, does not take the explicit microstructural configuration and the corresponding solid-fluid coupling into consideration. This work investigates how the microstructural configuration and material properties of porous materials constitute the macroscopic poroelastic material behavior described by the classical Biot's theory. We introduced an asymptotic based homogenization method to correlate the macro- and micro-mechanical behaviors of poroelastic materials, where an elastic solid and Newtonian fluid of low viscosity are considered. Through this homogenization process, the generalized Darcy's law, homogenized macroscopic continuity equation, and homogenized macroscopic equilibrium equation were obtained, where the homogenized macroscopic continuity and equilibrium equations reassemble the governing equations in Biot's theory.

For an effective modeling of microstructures, a numerical solution for PDEs based on a strong form collocation that employs image pixels as the discretization points is proposed. To achieve this objective, a gradient reproducing kernel collocation method (G-RKCM) formulated based on the partition of nullity and gradient reproducing conditions was developed. This approach reduces the order of differentiation to the first order when solving second order PDEs with strong form collocation. We showed that the same number of collocation points and source points can be used in G-RKCM for optimal convergence, unlike other strong form collocation methods. In addition, same order of convergence rate in the solution and its first order derivative are achieved, owing to the imposition of gradient reproducing conditions. The computational complexity of G-RKCM is also shown to be an enhancement over other strong form collocation methods, such as the reproducing kernel collocation method (RKCM).

In this work, we introduced the active contour model based on variational level set formulation for interface identification and boundary segmentation for the discretization of microstructures based on medical images. Using pixel point discretization, we introduced the RKCM and G-RKCM to solve the level set equation. In particular, the G-RKCM has been shown be effective since the second derivatives of the level set function involved in the regularization term are approximated by the first order differentiations of the gradient RK shape functions. We further showed that a B-spline kernel function with lower continuity can be preferably used to avoid the oscillation of level set functions in the two-color images.

The image based G-RKCM was applied to model trabecular bone microstructures with complex geometry for both solid and fluid phases. The corresponding numerical issues such as interface discretization and kernel function support size selection have been addressed. The investigation on the proper choice of unit cell dimension and image resolution has been performed, which provides guidance in the image-based trabecular bone modeling. The validation of the proposed image based multiscale modeling framework has been carried out by comparing the numerical prediction of effective material properties with experimental data of trabecular bone in the literature and solving a macroscopic trabecular bone problem using the homogenized material constants.

The numerical simulation of transient dynamic failure of structures subjected to blast loadings requires the key physics such as strong shocks in fluid (explosive gas and air) and solid media, fluid-structure interaction, material damage and fragmentations, and multi-body contact to be properly considered in the mathematical formulation and the associated numerical algorithms. These dominant phenomena in blast events yield “rough solution” in the conservation equations in the form of moving discontinuities that cannot be effectively modeled by the conventional finite element methods. A semi-Lagrangian meshfree Reproducing Kernel Particle Method (RKPM) framework is proposed to model such extreme events in this study. In this work, shock waves in both air and solid are modeled by embedding the Godunov flux into the semi-Lagrangian RKPM formulation in a unified manner. The essential shock physics are introduced in the proposed node-based Riemann solver, and the Gibbs oscillation is limited by introducing a gradient smoothing technique. In this thesis, two formulations are proposed and verified by solving a set of multi-dimensional benchmark problems involving strong shocks in fluids and solids. The air-structure interface is treated by a level set enhanced natural kernel contact algorithm, which does not require the definition of potential contact surfaces a priori. The blast-induced fragmentation is simulated by the damage model under the semi-Lagrangian RKPM discretization without using the artificial element erosion technique. Several benchmark problems have been solved to verify the accuracy and performance of the proposed numerical formulation. This computational framework is then applied to the simulation of a reinforced concrete column subjected to blast loading and explosive welding processes, demonstrating the effectiveness and robustness of the proposed methods.

Structures involving multiple materials are difficult for meshfree methods to model accurately due to the strain discontinuity introduced at the material interface. An immersed Reproducing Kernel Particle Method (RKPM) approach is proposed to model inhomogeneous materials using an immersed domain approach to allow independent approximations and discretizations for the background matrix and the foreground inclusion. In this approach, Nitsche’s method is introduced to enforce the interface compatibility conditions in a variationally consistent manner. The proposed method simplifies the spatial discretization procedures for multi-material problems involving complex geometries because the conforming requirements in discretization at the interface are avoided. Efficient and stable domain integration methods for the immersed RKPM discretization are investigated, and the performance of several approaches are compared. Specifically, conforming and non-conforming domain integration between the foreground and background domains are discussed. Optimal convergence is achieved without tedious procedures such as enrichment functions or boundary singular kernels commonly employed in other meshfree methods for solving multi-material problems. Several numerical examples are presented to examine the effectiveness of the proposed method. A non-linear formulation of the immersed RKPM method is also presented and its effectiveness in modeling brittle materials using an elastic-damage model is investigated.

In the extreme event such as air-blast or explosion, strong shocks lead to severe damage and fragmentation in structures. Despite the considerable effort made in recent years, reliable numerical prediction of fragmentation processes in materials and solids under blast loading or shock wave remains challenging. The conventional mesh-based methods (e.g., finite element method (FEM)) are ineffective due to large deformation-induced mesh distortion issues and exhibit non-convergent solutions in fracture problems. The meshfree method, such as reproducing kernel particle method (RKPM), naturally avoids computational difficulties associated with low-quality meshes, allows efficient adaptive refinement, and provides flexible control of smoothness and locality in numerical approximations.

The objective of this work is to develop a computational framework for effective modeling of shock dynamics in fluids and fluid-structure interactive systems. In this work, a stabilized RKPM framework for modeling shock waves in fluids is first developed. To capture essential shock physics and to avoid numerical oscillations, a Riemann-enriched smoothed flux divergence with an oscillation limiter is introduced under the stabilized conforming nodal integration (SCNI) framework. Besides, a flux splitting approach is employed to avoid advection-induced instabilities in fluid modeling, and the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL)-type oscillation limiter is employed to avoid over and undershooting of the numerical solution at shock front and to capture moving discontinuities with minimal diffusion. The proposed methods, termed MUSCL-SCNI, have been applied to the shock tube problem, compressible flow with vortex, and explosive detonation.

Next, an immersed RKPM formulation is developed for an effective body-unfitted spatial discretization of subdomains in heterogeneous materials and fluid structure interaction (FSI) problems involving complex geometries. RKPM naturally avoids computational challenges associated with low-quality meshes, allows efficient adaptive refinement, and provides flexible control of continuity and locality in the numerical approximations. A variational multiscale immersed method (VMIM) is proposed, where the solution fields are decoupled into coarse- and fine-scales, and the fine-scale solution represents the residual of the coarse-scale equations. Under VMIM, the coupling between different subdomains is done through a volumetric constraint, and the embedment of the fine-scale solution into coarse-scale equations yields a stabilized Galerkin formulation with enhanced stability and accuracy. The proposed method is first applied to modeling heterogeneous materials. It is then further extended to shock wave modeling in the FSI systems, where the meshfree algorithm based on MUSCL-SCNI is employed for ensured stability. Finally, the proposed VMIM is applied to air-blast events simulations.

The modeling of problems with singularities and discontinuities, such as cracks in elastic media, is one of the most challenging problems in computational mechanics. The numerical solution of such problems necessitates sufficient resolution, which is very costly. In this work, model order reduction (MOR) techniques are developed for problems with singularities based on meshfree methods. Theoretical investigation on the performance of the proposed methods via error estimation, stability analysis, and complexity analysis is also carried out in this research.

First, the full discrete model is constructed using enriched meshfree approximation to properly approximate singularities and discontinuities. This requires high order quadrature rules for accurate integration of enrichment functions and their derivative. An integrated singular basis function method (ISBFM) is introduced to circumvent this difficulty and the resulting ISBFM Galerkin formulation only integrates the enrichment functions on the boundaries away from the singularity point. This ISBFM Galerkin formulation also allows effective MOR procedures proposed in this work.

Two MOR methods via projection of a discrete system obtained from the ISBFM Galerkin formulation have been proposed, the ISBFM-UR method based on a uniform projection and the ISBFM-DR method based on a decomposed projection of smooth and non-smooth DOF in order to retain the singular behavior of the full scale solution.

Error estimation and stability analysis show that the reduced solution from ISBFM-DR can achieve a much-enhanced accuracy with slight increase of condition number compared to ISBFM-UR. While the stability analysis shows that the two MOR methods provide better conditioning than the full scale system, it also suggests that the conditioning of the ISBFM-DR method is compromised by its better accuracy compared to that of the ISBFM-UR method. Nevertheless, the complexity analysis shows that the ISBFM-DR method provides a better efficiency in addition to the enhanced accuracy compared to the ISBFM-UR method.

The proposed MOR methods are applied Poisson problems with singularities and LEFM problems. The numerical tests validate the effectiveness of the MOR methods, and the numerical results on accuracy and stability of the proposed methods are shown to be in good agreement with the analytical predictions carried in this research.

Achieving good accuracy while keeping a low computational cost in numerical simulations

of problems involving large deformations, material fragmentation and crack propagations still

remains a challenge in computational mechanics. For these classes of problems, meshfree

discretizations of local and nonlocal approaches, have been shown to be effective as they avoid

some of the common issues associated with mesh-based techniques, such as the need for re-

meshing due to excessive mesh distortion. Nonetheless, other issues remain.

In the framework of local mechanics, the semi-Lagrangian reproducing kernel particle

method (RKPM) has been proved to be particularly effectively for material damage and frag-

mentation, as by reconstructing the field approximations in the current configuration it does

not require the deformation gradient to be positive definite. This, however, results in a high

computational cost.

Furthermore, for crack propagation problems, the use of classical local mechanics presents

many challenges, such as the need of accurately representing the singular stress field at crack tips. The peridynamic nonlocal theory circumvents these issues by reformulating solid mechanics in terms of integral equations. In engineering applications, a simple node-based discretization of peridynamics is typically employed. This approach is limited to first order convergence and often lacks the symmetry of interaction of the continuous form. The latter can be recovered through the use of the peridynamic weak form, which however involves costly double integration.

First, we first propose, in the context of local mechanics, a blending-based spatial coupling

scheme to transition from the computationally cheaper Lagrangian RKPM to the semi-Lagrangian RKPM. Next, we introduce an RK approximation to the field variables in strong form peridynam-ics to increase the order of convergence of peridynamic numerical solutions. Then, we develop an efficient n-th order symmetrical variationally consistent nodal integration scheme for RK enhanced weak form peridynamics.

Lastly, we propose a Waveform Relaxation Newmark algorithm for time integration of

the semi-discrete systems arising from meshfree discretizations of local and nonlocal dynamics

problems. This scheme retains the unconditional stability of the implicit Newmark scheme with

the advantage of the lower computational cost of explicit time integration schemes.

Numerical examples demonstrate the effectiveness of the proposed approaches.

Physics-based numerical simulation remains challenging as the complexity of today’s high-fidelity models has dramatically increased. Model order reduction (MOR) and data-driven modeling, based on the emerging techniques of data learning and physical modeling, present a promising way to tackle the computational bottleneck related to the computational intensity and model complexity.

Nevertheless, MOR has proven to be significantly more difficult for parameterized mechanics systems that exhibit a wide variety of parameter-dependent nonlinear behaviors or that involve localized essential features. The first objective of this work is to develop robust, physics-preserving MOR methods. As constructing a low-dimensional MOR model can be considered as the hybrid data-physics approach, one can optimize it through a learning process using both data and physical models. As such, we first propose a MOR method based on decomposed reduced-order projections that well preserve the essential near-tip characteristic for fracture mechanics. Moreover, we develop an enhanced reduced-order basis to construct a low-dimensional subspace, deriving from a generalized manifold learning framework that allows the employment of local information in the data structure during the learning phase. This approach can yield a robust reduced-order model against noise and outliers and is well suited for parameterized nonlinear physical systems. Finally, a nonlinear MOR for a meshfree Galerkin formulation with the stabilized conforming nodal integration (SCNI) scheme is developed to yield a pure node based MOR that is particularly effective for hyper-reduction techniques. A numerical example of two-phase hyperelastic solid with perturbed loading conditions is used to validate the effectiveness of the proposed reduction method.

The second goal of the dissertation is to develop a robust data-driven computational framework, which provides an alternative to conventional scientific computing for complex materials. This framework aims at performing physical simulation by directly interacting with material data via machine learning procedures instead of employing phenomenological constitutive models, and especially addressing the robustness issue associated with noisy and scarce data. To this end, we propose to search data solutions from a locally reconstructed convex hull associated with the k-nearest neighbor points, which leads to robustness to noisy data and ensures convergence stability. The accuracy and robustness of the proposed data-driven approach are demonstrated in the modeling of linear and nonlinear elasticity problems. In addition, we present a preliminary result of data-driven modeling of biological tissue using material data collected from laboratory testing on heart valve tissue, showing the potential of data-driven simulation by integrating physical modeling and machine learning techniques.

Hydro-mechanical damage processes occur in numerous geological hazards and engineering applications. Despite considerable effort made in the past years, reliable numerical prediction of failure processes in multiphase porous media remains challenging. This is mainly due to the complexity of the involved multi-physical phenomena, as well as the ineffectiveness of conventional mesh-based methods (e.g., FEM) which suffer from large deformation-induced mesh distortion issues and exhibit non-convergent solutions for damage and fracture problems. The objective of this work is to develop a robust meshfree computational framework for effective modeling of hydro-mechanical damage processes. To this end, a fluid pressure projection method is employed in conjunction with the stabilized conforming nodal integration to achieve a stable reproducing kernel mixed formulation for poromechanics. Moreover, a damage particle method is developed for fracture modeling, where a smeared description of cracks is adopted to circumvent the burden associated with modeling complex crack patterns. To eliminate the pathological discretization size sensitivity, a scaling law is introduced in the damage model to ensure that the bulk damage energy dissipation over the nodal representative volume is consistent with the surface fracture energy of the crack segment. In addition, the smeared strain is computed as the boundary integral of displacements in each nodal representative domain, which avoids the ambiguity of taking direct derivatives of non-smooth displacements in the cracking region. As such, the computation of field and state variables along with the regularization procedure are all performed at nodal points, without any interpolation commonly needed in FEM. By incorporating the hydro-mechanical coupling, the damage particle method is capable of capturing the growth of fluid-driven cracks without tedious treatments of moving discontinuities. The effectiveness of the developed meshfree formulation is demonstrated in the modeling of hydraulic fracturing and landslides, which involves extreme deformation phenomena and interactions between solid, water and air phases.