Physics-Informed Neural Networks (PINNs) have emerged as a tool for approximating the solution of Partial Differential Equations (PDEs) in both forward and inverse problems. PINNs minimize a loss function which includes the PDE residual determined for a set of collocation points. Previous work has shown that the number and distribution of these collocation points have a significant influence on the accuracy of the PINN solution. Therefore, the effective placement of these collocation points is an active area of research. Specifically, available adaptive collocation point sampling methods have been reported to scale poorly in terms of computational cost when applied to high-dimensional problems. In this work, we address this issue and present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN). PACMANN incrementally moves collocation points toward regions of higher residuals using gradient-based optimization algorithms guided by the gradient of the PINN loss function, that is, the squared PDE residual. We apply PACMANN to several forward and inverse problems, including one with a low-regularity solution and 3D Navier Stokes, and demonstrate that this method matches the performance of state-of-the-art methods in terms of the accuracy/efficiency tradeoff for the low-dimensional problems, while outperforming available approaches for high-dimensional problems. Key features of the method include its low computational cost and simplicity of integration into existing physics-informed neural network pipelines. The code is available at https://github.com/CoenVisser/PACMANN.

A computational framework based on a Discontinuous Galerkin (DG)/Cohesive Zone formulation is utilized to simulate the experiments of the Purdue Damage Mechanics Modeling Challenge. The inelastic response of the additively-manufactured gypsum material used in the experimental tests is modeled via a dilatational plasticity model. The constitutive and fracture model parameters are calibrated using the load–displacement curves corresponding to three-point bending tests initially provided by the Challenge organizers. The test samples contained initial notches especially designed to force specific types of mixed fracture modes. The calibrated computational modeling framework is used to blindly simulate the more complex configuration of the Challenge experiments. The numerical predictions of the load–displacement curve and the shape of the curved fracture surface are compared to the experimental data provided a posteriori. It is found that the computational method is able to quantitatively describe the fracture response of the material including crack propagation, plastic wake, and the curved geometry of the fracture surface that results from the evolving fracture mode mixity with significant fidelity.

Slender beams are often employed as constituents in engineering materials and structures. Prior experiments on lattices of slender beams have highlighted their complex failure response, where the interplay between buckling and fracture plays a critical role. In this paper, we introduce a novel computational approach for modeling fracture in slender beams subjected to large deformations. We adopt a state-of-the-art geometrically exact Kirchhoff beam formulation to describe the finite deformations of beams in three-dimensions. We develop a discontinuous Galerkin finite element discretization of the beam governing equations, incorporating discontinuities in the position and tangent degrees of freedom at the inter-element boundaries of the finite elements. Before fracture initiation, we enforce compatibility of nodal positions and tangents weakly, via the exchange of variationally-consistent forces and moments at the interfaces between adjacent elements. At the onset of fracture, these forces and moments transition to cohesive laws modeling interface failure. We conduct a series of numerical tests to verify our computational framework against a set of benchmarks and we demonstrate its ability to capture the tensile and bending fracture modes in beams exhibiting large deformations. Finally, we present the validation of our framework against fracture experiments of dry spaghetti rods subjected to sudden relaxation of curvature.

The dynamic response of flexible filaments immersed in viscous fluids is important in cell mechanics, as well as other biological and industrial processes. In this paper, we propose a parallel computational framework to simulate the fluid-structure interactions in large assemblies of highly-flexible filaments immersed in a viscous fluid. We model the deformation of each filament in 3D with a C¹ geometrically-exact large-deformation finite-element beam formulation and we describe the hydrodynamic interactions by a boundary element discretization of the Stokeslet model. We incorporate a contact algorithm that prevents fiber interpenetration and avoids previously reported numerical instabilities in the flow, thus providing the ability to describe the complex evolution of large clouds of fibers over long time spans. In order to support the required long-term integration, we use implicit integration of the solid-fluid-contact coupling. We address the challenges associated with the solution of the large and dense linear system for the hydrodynamic interactions by taking advantage of the massive parallelization offered by Graphic Processing Units (GPUs), which we test up to 1000 fibers and 45000 degrees of freedom. We validate the framework against the well-established response of the sedimentation of a single fiber under gravity in the low to moderate flexibility range. We then reproduce previous results and provide additional insights in the large to extreme flexibility range. Finally, we apply the framework to the analysis of the sedimentation of large clouds of filaments under gravity, as a function of fiber flexibility. Owing to the long time spans afforded by our computational framework, our simulations reproduce the breakup response observed experimentally in the lower flexibility range and provide new insights into the breakup of the initial clouds in the higher flexibility range.

Intense surface eruptions are observed along the curved surface of a confined cylindrical film of hydrogel subject to laser-induced converging-diverging shock loading. Detailed numerical simulations are used to identify the dominant mechanisms causing mechanical instability. The mechanisms that produce surface instability are found to be fundamentally different from both acoustic parametric instability and shock-driven Richtmyer-Meshkov instability. The time scale of observed and simulated eruption formation is much larger than that of a single shock reflection, in stark contrast to previously studied shock-driven instabilities. Moreover, surface undulations are only found along external, as opposed to internal, soft solid boundaries. Specifically, classic bubble surface instability mechanisms do not occur in our experiments and here we comment only on the new surface undulations found along the outer boundary of solid hydrogel cylinders. Our findings indicate a new class of impulsively excited surface instability that is driven by cycles of internal shock reflections.

Modeling the spontaneous evolution of morphology in natural systems and its preservation by proportionate growth remains a major scientific challenge. Yet, it is conceivable that if the basic mechanisms of growth and the coupled kinetic laws that orchestrate their function are accounted for, a minimal theoretical model may exhibit similar growth behaviors. The ubiquity of surface growth, a mechanism by which material is added or removed on the boundaries of the body, has motivated the development of theoretical models, which can capture the diffusion-coupled kinetics that govern it. However, due to their complexity, application of these models has been limited to simplified geometries. In this paper, we tackle these complexities by developing a finite element framework to study the diffusion-coupled growth and morphogenesis of finite bodies formed on uniform and flat substrates. We find that in this simplified growth setting, the evolving body exhibits a sequence of distinct growth stages that are reminiscent of natural systems, and appear spontaneously without any externally imposed regulation or coordination. The computational framework developed in this work can serve as the basis for future models that are able to account for growth in arbitrary geometrical settings, and can shed light on the basic physical laws that orchestrate growth and morphogenesis in the natural world.

The propagation of cracks driven by a pressurized fluid emerges in several areas of engineering, including structural, geotechnical, and petroleum engineering. In this paper, we present a robust numerical framework to simulate fluid-driven fracture propagation that addresses the challenges emerging in the simulation of this complex coupled nonlinear hydro-mechanical response. We observe that the numerical difficulties stem from the strong nonlinearities present in the fluid equations as well as those associated with crack propagation, from the quasi-static nature of the problem, and from the a priori unknown and potentially intricate crack geometries that may arise. An additional challenge is the need for large scale simulation owing to the mesh resolution requirements and the expected 3D character of the problem in practical applications. To address these challenges we model crack propagation with a high-order hybrid discontinuous Galerkin/cohesive zone model framework, which has proven massive scalability properties, and we model the lubrication flow inside the propagating cracks using continuous finite elements, furnishing a fully-coupled discretization of the solid and fluid equations. We find that a conventional Newton–Raphson solution algorithm is robust even in the presence of crack propagation. The parallel approach for solving the linearized coupled problem consists of standard iterative solvers based on domain decomposition. The resulting computational approach provides the ability to conduct highly-resolved and quasi-static simulations of fluid-driven fracture propagation with unspecified crack path. We conduct a series of numerical tests to verify the computational framework against known analytical solutions in the toughness and viscosity dominated regimes and we demonstrate its performance in terms of robustness and parallel scalability, enabling simulations of several million degrees of freedom on hundreds of processors.

Recent experiments on hydrogels subjected to large elongations have shown elastic instabilities resulting in the formation of geometrically intricate fringe and fingering deformation patterns. In this paper, we present a robust numerical framework addressing the challenges that emerge in the simulation of this complex material response from the onset of instability to the post-bifurcation behavior. We observe that the numerical difficulties stem from the non-convexity of the strain energy density in the near-incompressible, large-deformation regime, which is responsible for the coexistence of multiple equilibrium paths with vastly-different, sinuous deformation patterns immediately after bifurcation. We show that these numerical challenges can be overcome by using sufficiently-high order of interpolation in the finite element approximation, an arc-length-based nonlinear solution procedure that follows the entire equilibrium path of the system, and an implementation enabling parallel, large-scale simulations. The resulting computational approach provides the ability to conduct highly-resolved, truly quasi-static simulations of complex elastic instabilities. We present numerical results illustrating the ability of the path-following approach to describe the full evolution of fringe and fingering instabilities observed experimentally in recent experiments of confined cylindrical specimens of soft hydrogels subject to tension. Importantly, we observe that the robustness of the static solution procedure enables complete access to the multiplicity of solutions occurring immediately after the onset of bifurcation, as well as to the settled post-bifurcation states.

We propose a mathematical model and a discretization strategy for the simulation of pressurized fractures in porous media accounting for the poroelastic effects due to the interaction of pressure and flow with rock deformations. The aim of the work is to develop a numerical scheme suitable to model the interplay among several fractures subject to fluid injection in different geometric configurations, in view of the application of this technique to hydraulic fracturing. The eXtended Finite Element Method, here employed for both the mechanical and fluid-dynamic problems, is particularly useful to analyze different configurations without remeshing. In particular, we adopt an ad hoc enrichment for the displacement at the fracture tip and a hybrid dimensional approach for the fluid. After the presentation of the model and discretization details we discuss some test cases to assess the impact of fracture spacing on aperture during injection.

The present work proposes a novel method for the simulation of crack propagation in brittle elastic materials that combines two of the most popular approaches in literature. A large scale displacement solution is obtained with the well known extended finite elements method (XFEM), while propagation is governed by the solution of a local phase field problem at the tip scale. The method, which we will refer to as Xfield, is here introduced and tested in 2D under mixed modes I and II loads. The main features and the capability of the Xfield to efficiently simulate crack propagation are shown in some numerical benchmarks.

We propose a mathematical model and a numerical scheme to describe compaction processes in a sedimentary rock layer undergoing both mechanical and geochemical processes. We simulate the sedimentation process by providing a sedimentation rate, and we account for chemical reactions using simplified kinetics describing either the conversion of a solid matrix into a fluid, as in the case of kerogen degradation into oil, or the precipitation of a mineral solute on the solid matrix of the rock. We use a Lagrangian description that enables to recast the equations in a fixed frame of reference. We present an iterative splitting scheme that allows solving the set of governing equations efficiently in a sequential manner. We assess the performances of this strategy in terms of convergence and mass conservation. Some numerical experiments show the capability of the scheme to treat two test cases, one concerning the precipitation of a mineral, the other the dissolution of kerogen.