A thermodynamically consistent constitutive model for fluid mud

In this contribution, an approach towards a thermodynamically consistent constitutive model for fluid mud is presented. Fluid mud exhibits highly non-Newtonian, thixotropic behaviour. It can be classified as a structured fluid. Typically, its viscosity is modeled using Bingham-type rheological models of different complexity [1, 2]. Here, the three-dimensional non-Newtonian constitutive behaviour will be modeled based on a visco-elasto-plastic model. At the current stage, a Drucker-Prager-like yield function has been formulated. Viscosity is assumed to be a function of shear viscosity. First results show the general ability to represent experimental data. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A thermodynamically consistent model of shape-memory alloys

This contribution presents a non-isothermal mesoscopic model of single-crystalline shape-memory alloys within the framework of continuum mechanics. We briefly recall static mesoscopic modeling concepts as presented in e.g. [4, 5] and propose a thermomechanically consistent model featuring the heat equation and thermo-mechanical coupling. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polyconvex potentials,invertible deformations,and thermodynamically consistent formulation of the nonlinear elasticity equations

It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that Ball’s approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.     

A temperature-based GENERIC approach for the thermodynamically consistent integration of thermoelastic solids

This work deals with the thermodynamically consistent (TC) time integration of thermoelastic systems with polyconvex density functions using the notion of the tensor-cross-product. While energy-momentum preserving integrators are well-known for conservative (isothermal) mechanical systems, Romero introduced in [7, 8] the new class of TC integrators. While [8] dealt with the sample application of thermo-elastodynamics, the scope of application was extended in [2] to coupled thermo-viscoelastodynamics in temperature form. A first step towards the systematic design of a TC integrator is to cast the evolution equations into the GENERIC (General Equation for Non-Equilibrium Reversible-Irreversible Coupling) framework [6] which reveals additional underlying physical structures of the system. Relying on a polyconvex density function and using the notion of the tensor-cross-product [1] we arrive at a polyconvex version of the GENERIC framework. Further applying the notion of a discrete gradient leads to a TC integrator. Using the entropy as the thermodynamical state variable as in [5, 8] the GENERIC framework possesses an easy structure. However, this choice of thermodynamical state variable only allows to prescribe entropy Dirichlet boundary conditions directly. This drawback can be compensated by using Lagrange-multipliers to be able to handle temperature Dirichlet boundary conditions leading to an extended system of algebraic equations to be solved, see [5]. Alternatively, the present work uses the temperature as the thermodynamical state variable, see also [2, 3] and the use of an energy-based Newton-Raphson termination criterion. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerically stable form of the simplex algorithm

Standard implementations of the Simplex method have been shown to be subject to computational instabilities, which in practice often result in failure to achieve a solution to a basically well-determined problem. A numerically stable form of the Simplex method is presented with storage requirements and computational efficiency comparable with those of the standard form. The method admits non-Simplex steps and this feature enables it to be readily generalized to quadratic and nonlinear programming. Although the principal concern in this paper is not with constraints having a large number of zero elements, all necessary modification formulae are given for the extension to these cases.     

A thermodynamically consistent numerical method for a phase field model of solidification

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.     

A linear energy and entropy-production-rate preserving scheme for thermodynamically consistent crystal growth models

We present a linear, second order, energy and entropy-production-rate preserving scheme for a thermodynamically consistent phase field model for dentritic crystal growth, combining an energy quadratization strategy with the finite element method. The scheme can be decomposed into a series of Poisson equations for efficient numerical implementations. Numerical tests are carried out to verify the accuracy of the scheme and simulations are conducted to demonstrate the effectiveness of the scheme on benchmark examples.     

A numerically stable algorithm for two server queue models

In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.     

A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example. This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.     

A consistent diffusion approximation for finite-capacity multiserver queues

A diffusion approximation is developed for general multiserver queues with finite waiting spaces, which are typical models of manufacturing systems as well as computer and communication systems. The model is the standard GI/G/s/s + r queue with s identical servers in parallel, r extra waiting spaces, and the first-come, first-served discipline. The main focus is on the steady-state distribution of the number of customers in the system. The process of the number of customers is approximated by a time-homogeneous diffusion process on a closed interval in the nonnegative real line. A conservation law plus some heuristics standing on solid theoretical ground generate approximation formulas for the steady-state distribution and other congestion measures. These formulas are consistent with the exact results for the M/G/s/s and M/M/s/s + r queues. The accuracy of approximations for principal congestion measures are numerically examined for some particular cases.     

A numerically stable and efficient technique for the maintenance of positive definiteness in the Hessian for Newton-type methods

Unconstrained optimization problems using Newton-type methods sometimes require that the Hessian matrix, G, calculated at each iteration, be modified to G¹ in order to insure that the direction of search is downhill. It is shown that several previously proposed methods modify G in such a manner that G¹ becomes extremely ill-conditioned even when G itself is well conditioned. The method proposed here is a modification of Greenstadt's, where bounds on the eigenvalues of G¹ may be imposed such that G¹ has a spectral condition number identical to G when G is well-conditioned but indefinite. The modification updates G by the addition of rank-one matrices, which are obtained by a partial eigenvalue decomposition of G, rather than a complete one as originally proposed by Greenstadt. The matrix G¹ obtained in this manner is identical to the G¹ obtained by Greenstadt's method, but may be computed in substantially less time.     

A novel class of thermodynamically consistent isotropic hardening rules for some damage models at finite strains

This paper proposes isotropic hardening rules for plasticity damage models which satisfy the 2nd law of thermodynamics. Here, the Gurson -model modified by Tvergaard/Needleman is used as a prototype model for finite strains to discuss main properties of the novel hardening rules. Additionally it is shown that thermodynamically consistent rules are suitable to fit experimental data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerically stable least squares solution to the quadratic programming problem

The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.      

A consistent implementation for inelastic materials in an ALE formulation for steady state rolling contact

The numerical analysis of rolling contact for rubber materials is a challenging task, especially due to the many nonlinearities inherent to the material, large deformations, friction, and energy dissipation, among others. Industrial applications can be found in ball bearings, rollers, and most commonly in tires of vehicles, applications where reliable numerical simulations lead to the improvement of durability, performance and safety. While a transient analysis stands as a practical and powerful tool for the simulation of rotating bodies, the large amount of computational resources required represents its biggest disadvantage. An alternative frequently used lays in a steady state simulation by means of an Arbitrary Lagrangian Eulerian (ALE) formulation, where the rotational velocity and axial loads are assumed to remain constant. Within this framework, the reference configuration is neither attached to the material particles nor fixed in space and special attention should be paid to the history variables of inelastic materials. In this work, a viscoelastic material model is implemented in an in-house finite element code, based on a generalized Maxwell model. The implementation takes into consideration the contribution of all elements connected in circumferential direction and a consistent linearization is made for each of them, leading to an assembled stiffness matrix with more non-zero values than a standard one. This approach is combined with smeared reinforcement embedded in base elements. The reinforcing layers are described by a hyperelastic material model, providing additional advantages for the modeling and simulation of reinforced rollers and tires. Numerical results for different examples show the capabilities of this implementation and the efficiency of the numerical algorithms is discussed. Important remarks and an outlook for further research concludes this presentation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of the formulation for eddy diffusion in two one-dimensional stratification models

Two models for thermal stratification based on turbulent diffusion concepts are analysed and compared. The models by Henderson-Sellers; and by McCormick and Scavia, are shown to be equivalent at large values of the Richardson number, R_i. At small R_i, the simpler model reverts to specification of the turbulent diffusion as a constant value. This simplification is also demonstrated to be a realistic approximation only at low wind speeds and for deep lakes. By comparison of these model types, a (previously empirically defined by McCormick and Scavia) parameter β is related conceptually to the lake depth, H.     

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order , where is the machine precision, the new method computes the eigenvalues to full possible accuracy. Received April 8, 1996 / Revised version received December 20, 1996     

Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body‐implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi‐static approximation to the dynamics. We focus on the linearized system about stress‐free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time‐dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non‐stress free equilibria and states with residual stress and derive an energy law for the corresponding time‐dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement‐traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two‐dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimation of numerically stable step-size for neutral delay-differential equations via spectral radius

We derive the estimates of numerically stable step-size for systems of neutral delay-differential equations (NDDEs), which only need to be calculated the spectral radius of the corresponding matrices. The stable step-size for numerical integration of NDDEs can be easily selected by means of the estimates. The stability regions of both linear multistep methods and explicit Runge-Kutta methods are presented.     

A volume‐consistent discrete formulation of aggregation population balance equations

In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non‐uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher‐order moments as well as number density distribution with high accuracy. Copyright © 2015 John Wiley & Sons, Ltd.