Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws

This paper reports a theoretical and numerical framework to model nonlinear waves in elastic-plastic solids. Formulated in the Eulerian frame, the governing equations employed include the continuity equation, the momentum equation, and an elastic-plastic constitutive relation. The complete governing equations are a set of first-order, fully coupled partial differential equations with source terms. The primary unknowns are velocities and deviatoric stresses. By casting the governing equations into a vector-matrix form, we derive the eigenvalues of the Jacobian matrix to show the wave speeds. The eigenvalues are also used to calculate the Courant number for numerical stability. The model equations are solved using the Space-Time Conservation Element and Solution Element (CESE) method. The approach is validated by comparing our numerical results to an analytical solution for the special case of longitudinal wave motion.Comment: 34 pages, 11 figure

Finite volume modelling of low speed structural impact problems

Two investigations are described in this thesis on the common theme of applying finite volume methods to simulate structural impact problems. The first investigation is the application of the Eulerian Finite Volume Method (EFVM) to simulate the low-speed impact of ductile materials. Simulation results are validated against experiment showing that it is possible to accurately predict crater deformation profiles over the low speed speed impact regime for different projectile and substrate materials. We demonstrate how the rate dependent Johnson-Cook plasticity model is crucial to ensure correspondence to experiment. The second investigation is concerned with the application of EFVM to simulate impact damage to thin polymeric coatings applied to the surface of metals. The aim of this work is to demonstrate how new simulation methods can help understand coating damage due stone impact. We simulate the debonding phenomenon of single layer coatings under impact by setting boundary conditions at the plate and paint interface. We show how EFVM can capture two limits of interface behaviour, sliding and separation 'slip' at one extreme and zero sliding 'welded' at the other. Results compare well to previously published experimental and simulation work, and our own finite element simulations in Abaqus. We also demonstrate how EFVM brings greater robustness and stability compared to FEM when modelling adhesive failure and higher energy impact penetration

Enhanced SPH modeling of free-surface flows with large deformations

The subject of the present thesis is the development of a numerical solver to study the violent interaction of marine flows with rigid structures. Among the many numerical models available, the Smoothed Particle Hydrodynamics (SPH) has been chosen as it proved appropriate in dealing with violent free-surface flows. Due to its Lagrangian and meshless character it can naturally handle breaking waves and fragmentation that generally are not easily treated by standard methods. On the other hand, some consolidated features of mesh-based methods, such as the solid boundary treatment, still remain unsolved issues in the SPH context. In the present work a great part of the research activity has been devoted to tackle some of the bottlenecks of the method. Firstly, an enhanced SPH model, called delta-SPH, has been proposed. In this model, a proper numerical diffusive term has been added in the continuity equation in order to remove the spurious numerical noise in the pressure field which typically affects the weakly-compressible SPH models. Then, particular attention has been paid to the development of suitable techniques for the enforcement of the boundary conditions. As for the free-surface, a specific algorithm has been designed to detect free-surface particles and to define a related level-set function with two main targets: to allow the imposition of peculiar conditions on the free-surface and to analyse and visualize more easily the simulation outcome (especially in 3D cases). Concerning the solid boundary treatment, much effort has been spent to devise new techniques for handling generic body geometries with an adequate accuracy in both 2D and 3D problems. Two different techniques have been described: in the first one the standard ghost fluid method has been extended in order to treat complex solid geometries. Both free-slip and no-slip boundary conditions have been implemented, the latter being a quite complex matter in the SPH context. The proposed boundary treatment proved to be robust and accurate in evaluating local and global loads, though it is not easy to extend to generic 3D surfaces. The second technique has been adopted for these cases. Such a technique has been developed in the context of Riemann-SPH methods and in the present work is reformulated in the context of the standard SPH scheme. The method proved to be robust in treating complex 3D solid surfaces though less accurate than the former. Finally, an algorithm to correctly initialize the SPH simulation in the case of generic geometries has been described. It forces a resettlement of the fluid particles to achieve a regular and uniform spacing even in complex configurations. This pre-processing procedure avoids the generation of spurious currents due to local defects in the particle distribution at the beginning of the simulation. The delta-SPH model has been validated against several problems concerning fluid-structure interactions. Firstly, the capability of the solver in dealing with water impacts has been tested by simulating a jet impinging on a flat plate and a dam-break flow against a vertical wall. In this cases, the accuracy in the prediction of local loads and of the pressure field have been the main focus. Then, the viscous flow around a cylinder, in both steady and unsteady conditions, has been simulated comparing the results with reference solutions. Finally, the generation and propagation of 2D gravity waves has been simulated. Several regimes of propagation have been tested and the results compared against a potential flow solver. The developed numerical solver has been applied to several cases of free-surface flows striking rigid structures and to the problem of the generation and evolution of ship generated waves. In the former case, the robustness of the solver has been challenged by simulating 2D and 3D water impacts against complex solid surfaces. The numerical outcome have been compared with analytical solutions, experimental data and other numerical results and the limits of the model have been discussed. As for the ship generated waves, the problem has been firstly studied within the 2D+t approximation, focusing on the occurrence and features of the breaking bow waves. Then, a dedicated 3D SPH parallel solver has been developed to tackle the simulation of the entire ship in constant forward motion. This simulation is quite demanding in terms of complexities of the boundary geometry and computational resources required. The wave pattern obtained has been compared against experimental data and results from other numerical methods, showing in both the cases a fair and promising agreement

Mesoscale modeling and direct simulation of explosively dispersed granular materials

Explosively dispersed granular materials frequently exhibit macroscale coherent particle clustering and jetting structures. The underlying mechanism is of significant interest to study instability and mixing in high-speed gas-solid flows but remains unclear, primarily attributed to the complex mesoscale multiphase interactions involved in the dispersal process. In order to advance the understanding of particle clustering and jetting instabilities, this thesis establishes a numerical framework for solving interface-resolved gas-solid flows with non-deforming bodies that are able to move, contact, and collide. The developed framework is implemented to create a computational solver and then verified using a variety of gas-solid flow problems at different geometric scales. Employing the developed framework and solver, this thesis further studies the particle clustering and jetting instabilities in explosively dispersed granular materials. A Cartesian, 3D, high-resolution, parallelized, gas-solid flow solver is created with the capability of tackling shocked flow conditions, irregular and moving geometries, and multibody collisions. The underlying numerical framework integrates operator splitting for partitioned fluid-solid interaction in the time domain, 2nd/3rd order strong stability-preserving Runge--Kutta methods and 3rd/5th order weighted essentially nonoscillatory schemes for high-resolution tempo-spatial discretization, the front-tracking method for evolving phase interfaces, a new field function developed for facilitating the solution of complex and dynamic fluid-solid systems on Cartesian grids, a new collision model developed for deterministic multibody contact and collision with parameterized coefficients of restitution and friction, and a new immersed boundary method developed for treating arbitrarily irregular and moving boundaries. The developed framework and solver are able to accurately, efficiently, and robustly solve coupled fluid-fluid, fluid-solid, and solid-solid interactions with flow conditions ranging from subsonic to hypersonic states. Employing the developed framework and solver, direct simulations that capture interface-resolved multiphase interactions and deterministic mesoscale granular dynamics are conducted to investigate particle clustering and jetting instabilities. A random sampling algorithm is employed to generate stochastic payload morphologies with randomly distributed particle positions and sizes. Through solving and analyzing cases that cover a set of stochastic payloads, burster states, and coefficients of restitution, a valid statistical dissipative property of the framework in solving explosively dispersed granular materials with respect to Gurney velocity is demonstrated. The predicted surface expansion velocities can extend the time range of the velocity scaling law with regard to Gurney energy in the Gurney theory from the steady-state termination phase to the unsteady evolution phase. When considering the mean surface expansion velocities, the maximum error of the unsteady velocity scaling law is about $0.792\%$ among the investigated Gurney energies. In addition, a dissipation analysis of the current discrete modeling of granular payloads suggests that incorporating the effects of porosity can enhance the prediction of Gurney velocity for explosively dispersed granular payloads. On the basis of direct simulations, an explanation for particle clustering and jetting instabilities is proposed to increase the understanding of established experimental observations in the literature. Results suggest that the development of internal sliding and colliding lines in the shock-compacted granular payload can be critical to the subsequent fracture pattern of the payload. Particle clusters manifested through payload fracture are then maintained by local pressure gradient between surrounding and interstitial flows as well as by dissipative inter-grain collisions. The existence of stable clusters introduce a more non-equilibrium momentum distribution in the overall payload, exhibiting as a form of clustering instability. Under the current assumptions of non-deformable grains, the mesoscale granular dynamics largely depends on the payload morphology as a result of packing methods. Different payload morphologies can develop varied sliding and colliding lines, which lead to a corresponding pattern for payload fracturing and particle clustering. With the rapid development of high-performance computing technology, future direct simulations on stochastic payloads with significantly increased domain sizes, number of particles, and solution times are expected to lead to a better understanding of the flow instability in explosively dispersed granular payloads. It is suggested that statistics collected from a large number of mesoscale computations based on random payload morphologies can potentially evolve into a macroscopic theory of multiphase flow instability for particle clustering and jetting phenomena widely observed in many areas involving dense gas-solid flows

Numerical Simulations of Bubble Dynamics Near Viscoelastic Media

Cavitation-induced damage occurs in a wide range of applications, including in naval hydrodynamics, medicine, and the Spallation Neutron Source. Local transient pressure decreases in liquid flow may give rise to explosive bubble growth and violent collapse, with shock waves produced at collapse interacting with neighboring solids. Although the mechanisms of erosion to hard, metallic solids can be predicted in relatively simple geometries, damage to soft materials (e.g., elastomeric coatings, soft tissue) or in confined geometries is less well understood. In such problems, the constitutive models describing the medium are non-trivial and include effects such as (nonlinear) elasticity, history (relaxation effects) and viscosity. As a result, the influence of the shock on the material and the response of the material to the shock are poorly understood. To gain fundamental insights into cavitation-induced damage to both soft objects and rigid materials, we develop a novel Eulerian approach for numerical simulations of wave propagation in heterogeneous viscoelastic media. We extend the five-equations multiphase, interface-capturing model, based on the idea that all the materials (gases, liquids, solids) obey the same equation of state with spatially varying properties, to incorporate the desired constitutive relation. We consider problems in which the deformations are small, such that the substances can be described by linear viscoelastic constitutive relations. One difficulty is the calculation of strains in an Eulerian framework, which we address by using a hypoelastic model in which an objective time derivative (Lie derivative) of the constitutive relation is taken to evolve strain rates. The resulting numerical framework is a solution-adaptive, high-order interface-capturing approach for compressible, multiphase flows involving linear viscoelasticity at all speeds. We then utilize this numerical framework to gain fundamental insights into cavitation damage (i) in a confined geometry, (ii) in shock wave lithotripsy, and (iii) to rigid objects covered by an elastomeric coating. We examine the maximum stresses, pressures, and temperatures along/in rigid/compliant objects. We quantify the effect of confinement on an inertially collapsing bubble and determine the appropriate scaling governing the maximum pressures can predicted on the surfaces. We investigate the stresses produced to a model kidney stone due to a shock wave by examining the amplification of tensile stresses in the stone when a gas bubble is present. The impact loads on a polymeric coating relevant to naval engineering applications by shock-induced bubble collapse indicate how pitting and coating material ejection may take place by repeated cavitation events near the surface.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147536/1/mrdz_1.pd