A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part I: The variational formulation for a single evolution equation

The objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow, fluid flow and species transport processes with sharply defined phase-change interfaces. In this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non-linear time-dependent partial differential equations. In Part II of this paper¹ we extend the formulation given here for a single evolution equation to a system of coupled evolution equations. In Part III² we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part III: Numerical implementation and computational results

In Part I of this paper,¹ the conceptual framework of a rate variational least squares formulation of a continuously deforming mixed-variable finite element method was presented for solving a single evolution equation. In Part II² a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed-variable least squares rate variational finite element method. The system of evolution equations describes the coupled heat flow, fluid flow and trace species transport in porous media under conditions when the flow velocities and constituent phase transitions induce sharp fronts in the solution domain. In this paper, we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non-linear ordinary differential equations. Next, we present computational results obtained using the code in which this deforming mixed finite element method was implemented. Because several features of the formulation are novel and have not been previously attempted, the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions.     

A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems

The finite volume based numerical approach is used to simulate phase-change processes including natural convection. This approach is based on a cell-by-cell, thermally driven mushy cell tracking equation, developed in Part I [20], to trace the front at which phase-change occurs. A mushy cell is a specialized cell where the interface between liquid and solid phases is located. In this paper, the mushy cell tracking equation and the associated boundary condition around the mushy cells are derived in a general manner and shown to have the same form as that used in Part I. The SIMPLE algorithm is adopted to solve the flow, including pressure field, as well, in the liquid phase and a conjugate gradient method is used when solving the system of discretized equations. To reduce computational time, an acceleration technique, based on a justified quasi-steady state assumption, is adopted. The proposed numerical method is applied to simulate the solidification and melting of Tin with natural convection. The numerical predictions are compared well with the available experimental data and previously published numerical results. Specifically, these comparisons demonstrate that the proposed methodology is capable of predicating the location of moving fronts and the temperature distributions for phase-change processes with natural convection.     

A hybrid extended finite element/level set method for modeling phase transformations

A hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented. We focus attention on two‐dimensional solidification problems, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X‐FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X‐FEM is suggested for this class of problems whereby the partition of unity is constructed with C¹(Ω) polynomials and enriched with a C⁰(Ω) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to significantly improve estimates for the front velocity. Temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a finer finite difference grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation. Copyright © 2002 John Wiley & Sons, Ltd.     

Free and Forced Vibrations of a Segmented Bar by a Meshless Local Petrov–Galerkin (MLPG) Formulation

We use the meshless local Bubnov–Galerkin (MLPG6) formulation to analyze free and forced vibrations of a segmented bar. Three different techniques are employed to satisfy the continuity of the axial stress at the interface between two materials: Lagrange multipliers, jump functions, and modified moving least square basis functions with discontinuous derivatives. The essential boundary conditions are satisfied in all cases by the method of Lagrange multipliers. The related mixed semidiscrete formulations are shown to be stable, and optimal in the sense that the ellipticity and the inf-sup (Babuška-Brezzi) conditions are satisfied. Numerical results obtained for a bimaterial bar are compared with those from the analytical, and the finite element methods. The monotonic convergence of first two natural frequencies, first three mode shapes, and a static solution in the L ², and H ¹ norms is shown. The relative error in the numerical solution for a transient problem is also very small.     

A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems

A thermally driven mushy cell tracking algorithm for phase-change problems with a moving boundary is presented. The equation used to track the moving boundary is based on energy balance over the mushy cell and is applied to advance a moving front in a cell-by-cell manner. The efficacy of the tracking algorithm is demonstrated on specific problems solved using the finite volume method. An implicit scheme is adopted to ensure that the numerical solution is unconditionally stable in time. A preconditioned conjugated gradient (P-CG) solver is implemented to ensure that solutions converge in a finite number of iterations. Four benchmark cases are used to validate the algorithm including solidification in one dimensional space (two-region problem), melting of pure aluminum in two-dimensional (2D) space, solidification with periodic boundary conditions, and solidification of one-region problem. The results obtained show that the current algorithm is capable of converging to accurate solutions for moving fronts and the numerical predications are in excellent agreement with corresponding analytical solutions.     

Adjoint-based optimal variable stiffness mesh deformation strategy based on bi-elliptic equations

There are many recent advances in mesh deformation methods for computational fluid dynamics simulation in deforming geometries. We present a method of constructing dynamic mesh around deforming objects by solving the bi-elliptic equation, an extension of the biharmonic equation. We show that introducing a stiffness coefficient field a(x) in the bi-elliptic equation can enable mesh deformation for very large boundary movements. An indicator of the mesh quality is constructed as an objective function of a numerical optimization procedure to find the best stiffness coefficient field a(x). The optimization is efficiently solved using steepest descent along adjoint-based, integrated Sobolev gradients. A multiscenario optimization procedure is performed to calculate the optimal stiffness coefficient field a^蜧(x) for a priori unpredictable boundary movements. Copyright © 2011 John Wiley & Sons, Ltd.     

Kinetics of low-temperature discontinuous deformation of metals

Kinetic characteristics of discontinuous yielding at a temperature of 4 K as functions of a number of factors are obtained using numerical simulation and experimental data for austenitic steel and aluminum alloy. During the development of a strain jump, the deformation rate and acceleration are 19 s⁻¹ and 5000 s⁻², respectively, for steel specimens and are much lower for aluminum alloy. The jump duration is mainly determined by the characteristics of the loading system. An equation relating the strain jump and the critical stress for low-temperature ductile materials is derived. The energy balance and the mechanism of low-temperature discontinuous yielding of metals are discussed. Its dynamic and thermally activated components are estimated taking into account the strain hardening of the material.     

A high‐order hybridizable discontinuous Galerkin method for elliptic interface problems

We present a high‐order hybridizable discontinuous Galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface. The hybridizable discontinuous Galerkin method is endowed with several distinct characteristics. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the global degrees of freedom. Second, they provide, for elliptic problems with polygonal interfaces, approximations of all the variables that converge with the optimal order of k + 1 in the L²(Ω)‐norm where k denotes the polynomial order of the approximation spaces. Third, they possess some superconvergence properties that allow the use of an inexpensive element‐by‐element postprocessing to compute a new approximate solution that converges with order k + 2. However, for elliptic problems with finite jumps in the solution across the curvilinear interface, the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric. The discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence. To recover the optimal convergence for the approximate solution and gradient, we propose to use superparametric elements at the interface. Copyright © 2012 John Wiley & Sons, Ltd.     

A probabilistic theory for the strength of discontinuous fibre composites

The strength of discontinuous fibre-reinforced composites is often reduced due to local stress concentrations at large fibre-end-gaps. A theoretical prediction of the strength of unidirectional fibre composites is performed based upon a probabilistic model of the fibre configuration. This work further develops the concepts of Bader, Chou and Quigley, and Fukuda and Chou. A limiting case of the present analysis shows good agreement with the results of Smith. Emphases are placed on the effect of matrix stress transfer properties including matrix plasticity. For a matrix deforming elastically, the strength is reduced as the composite size (N) increases. As compared with the rule-of-mixtures prediction for continuous fibre composites with identical fibre volume fraction, the reduction is shown to be proportional to (In N)^–P , with the exponent P being between 0.5 and 1 for two-dimensional composites and between 0.25 and 0.5 for three-dimensional composites. For a matrix deforming plastically, the local stress concentrations are reduced. Based upon the analytical expression of the local load sharing rule for a plastically deformed matrix, the composite strength is shown to approach the modified rule-of-mixtures of Kelly and Tyson as the matrix yield stress decreases.This work was done on leave from Applied Mechanics Section, Central Research Laboratories, Sumitomo Metal Industries, Ltd., Nishinagasuhumdori, Amagasaki, Japan.     

A combined FIC‐TDG finite element approach for the numerical solution of coupled advection–diffusion–reaction equations with application to a bioregulatory model for bone fracture healing

Numerical schemes for the approximative solution of advection–diffusion–reaction equations are often flawed because of spurious oscillations, caused by steep gradients or dominant advection or reaction. In addition, for strong coupled nonlinear processes, which may be described by a set of hyperbolic PDEs, established time stepping schemes lack either accuracy or stability to provide a reliable solution. In this contribution, an advanced numerical scheme for this class of problems is suggested by combining sophisticated stabilization techniques, namely the finite calculus (FIC‐FEM) scheme introduced by Oñate et al. with time‐discontinuous Galerkin (TDG) methods. Whereas the former one provides a stabilization technique for the numerical treatment of steep gradients for advection‐dominated problems, the latter ensures reliable solutions with regard to the temporal evolution. A brief theoretical outline on the superior behavior of both approaches will be presented and underlined with related computational tests. The performance of the suggested FIC‐TDG finite element approach will be discussed exemplarily on a bioregulatory model for bone fracture healing proposed by Geris et al., which consists of at least 12 coupled hyperbolic evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

This work addresses the numerical approximation of solutions to a dimensionless form of the Weertman equation, which models a steadily moving dislocation and is an important extension (with advection term) of the celebrated Peierls‐Nabarro equation for a static dislocation. It belongs to the class of nonlinear reaction‐advection‐diffusion integro‐differential equations with Cauchy‐type kernel, thus involving an integration over an unbounded domain. In the Weertman problem, the unknowns are the shape of the core of the dislocation and the dislocation velocity. The proposed numerical method rests on a time‐dependent formulation that admits the Weertman equation as its long‐time limit. Key features are (1) time iterations are conducted by means of a new, robust, and inexpensive Preconditioned Collocation Scheme in the Fourier domain, which allows for explicit time evolution but amounts to implicit time integration, thus allowing for large time steps; (2) as the integration over the unbounded domain induces a solution with slowly decaying tails of important influence on the overall dislocation shape, the action of the operators at play is evaluated with exact asymptotic estimates of the tails, combined with discrete Fourier transform operations on a finite computational box of size L; (3) a specific device is developed to compute the moving solution in a comoving frame, to minimize the effects of the finite‐box approximation. Applications illustrate the efficiency of the approach for different types of nonlinearities, with systematic assessment of numerical errors. Converged numerical results are found insensitive to the time step, and scaling laws for the combined dependence of the numerical error with respect to L and to the spatial step size are obtained. The method proves fast and accurate and could be applied to a wide variety of equations with moving fronts as solutions; notably, Weertman‐type equations with the Cauchy‐type kernel replaced by a fractional Laplacian.     

NUMERICAL SOLUTION OF TRANSIENT,FREE SURFACE PROBLEMS IN POROUS MEDIA

The focus of this paper is the development of numerical schemes for tracking the moving fluid surface during the filling of a porous medium (e.g., polymer injection into a porous mold cavity). Performing a mass balance calculation on an arbitrarily deforming control volume, leads to a general governing filling equation. From this equation, a general, fully time implicit, numerical scheme based on a finite volume space discretization is derived. Two numerical schemes are developed: (1) a fully deforming grid scheme, which explicitly tracks the location of the filling front, and (2) a fixed grid scheme, that employs an auxiliary variable to locate the front. The validity of the two schemes is demonstrated by solving a variety of one- and two-dimensional problems; both approaches provide predictions with similar accuracy and agree well with available analytical solutions.     

A Jacobian‐free‐based IIM for incompressible flows involving moving interfaces with Dirichlet boundary conditions

In this paper, a finite difference marker‐and‐cell (MAC) scheme is presented for the steady Stokes equations with moving interfaces and Dirichlet boundary condition. The moving interfaces are represented by Lagrangian control points and their position is updated implicitly using a Jacobian‐free approach within each time step. The forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions. The proposed Jacobian‐free Newton–generalized minimum residual (GMRES) method avoids the need to form and store the matrix explicitly in the computation of the inverse of the Jacobian and betters numerical stability. The Stokes equations are discretized on a MAC grid via a second‐order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient Uzawa‐type method. Numerical results demonstrate very well the accuracy and effectiveness of the proposed method. The present algorithm has been applied to solve incompressible Navier–Stokes flows with moving interfaces. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamics of micromechanisms controlling the mechanical behaviour of industrial single crystal superalloys

When deforming bulk material, micromechanisms involving moving defects result in mechanical characteristics observed at a macroscopic scale.In situ straining of microsamples in a Transmission Electron Microscope, provides the unique advantage of observing the dislocation dynamics involved in such micro-deformation processes under the combined effects of stress and temperature. Here the efficiency of this technique is illustrated by describing the different obstacles controlling the movement of dislocations in a two-phase industrial single crystal superalloy. At 25‡ and 850‡C, different core structures of the moving dislocations as well as several ways of crossing obstacles are described, which concern the movement of dislocations inγ channels, at γ/γ′ interfaces and while shearing γ′ precipitates. From these observations, a quantitative analysis is developed leading to the evaluation of the critical propagation stresses involved in the channels of the matrix and when crossing the interfaces. This allows to discuss the various sites of resistance opposed to the dislocation movements and controlling the macroscopic deformation.     

A least-squares front-tracking finite element method analysis of phase change with natural convection

This paper focuses on the numerical modelling of phase-change processes with natural convection. In particular, two-dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least-squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase-change problems.     

Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Eulerian frame of reference using ρ, u,p primitive variables

This paper presents a space–time least squares finite element formulation of one‐dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, p as primitive variables with C¹¹ type p‐version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C¹¹ type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time accurate numerical simulations show resolution of shock structure (i.e. shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as a model problem. True time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed non‐weak solutions have characteristics as those of the strong solutions of the gas dynamics equations. Copyright © 2001 John Wiley & Sons, Ltd.     

A fixed-grid numerical modelling of transient liquid phase bonding and other diffusion-controlled phase changes

A transient liquid phase (TLP), in which a liquid layer is formed and subsequently solidifies, and other diffusion-controlled phase changes are generally associated with moving phase-change interfaces. Both fixed and variable grid discretization models have been formulated to investigate these diffusion-controlled problems. However, all numerical efforts to date have employed one of the approaches explicitly to track the moving interfaces across which there exist step changes in concentrations. In this article, the fixed-grid source-based method originally developed to simulate the temperature fields for melting-solidification phase change processes has been adopted to simulate diffusion-controlled dissolution and solidification. This method solves a unique diffusion equation for the different phases and the moving interfaces using implicit time integration. Compared with previously developed models, it is not only simpler in numerical formulation and procedure, but also more convenient to extend to many phases and high-dimensional problems. We report here the detailed formulation of the relevant equations, and compare and validate the model using experimental data and previous modelling predictions for several systems available from the existing literature.     

Efficient Strip-Mode SAR Raw-Data Simulator of Extended Scenes Included Moving Targets Based on Reversion of Series

The Synthetic Aperture Radar (SAR) raw data generator is required to the evaluation of focusing algorithms, moving target analysis, and hardware design. The time-domain SAR simulator can generate the accurate raw data but it needs much time. The frequency-domain simulator not only increases the efficiency but also considers the trajectory deviations of the radar. In addition, the raw signal of the extended scene included static and moving targets can be generated by some frequency-domain simulators. However, the existing simulators concentrate on the raw signal simulation of the static extended scene and moving targets at uniform speed mostly. As for the issue, the two-dimensional signal spectrum of moving targets with constant acceleration can be derived accurately based on the geometric model of a side-looking SAR and reversion of series. And a frequency-domain algorithm for SAR echo signal simulation is presented based on the two-dimensional signal spectrum. The raw data generated with proposed method is verified by several simulation experiments. In addition to reveal the efficiency of the presented frequency-domain SAR scene simulator, the computational complexity of the proposed method is compared with the time-domain approach using the complex multiplication. Numerical results demonstrate that the present method can reduce the computational time significantly without accuracy loss while simulating SAR raw data.