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This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by Peshkov and Romenski [110] , further denoted as HPR model . In that framework, the viscous stresses are computed from the so-called distortion tensor A , which is one of the primary state variables in the proposed first order system. A very important key feature of the HPR model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids . Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of A . Also heat conduction is included via a first order hyperbolic system for the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the first and the second principle of thermodynamics. It is also fundamentally different from first order Maxwell–Cattaneo-type relaxation models based on extended irreversible thermodynamics. The HPR model represents therefore a novel and unified description of continuum mechanics, which applies at the same time to fluid mechanics and solid mechanics . In this paper, the direct connection between the HPR model and the classical hyperbolic–parabolic Navier–Stokes–Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit. From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER–WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemes to the HPR model in the stiff relaxation limit, and compare the numerical results with exact or numerical reference solutions obtained for the Euler and Navier–Stokes equations. Numerical convergence results are also provided. To show the universality of the HPR model, the paper is rounded-off with an application to wave propagation in elastic solids, for which one only needs to switch off the strain relaxation source term in the governing PDE system. We provide various examples showing that for the purpose of flow visualization , the distortion tensor A seems to be particularly useful. [ABSTRACT FROM AUTHOR]

• We have designed a generalized multiscale model for heterogeneous Cosserat media without scale separation. • We have designed multiscale spaces with and without coupling of processes and investigated them. • Numerical results are presented that show the proposed methods provide a good accuracy with a few degrees of freedom. In this paper, we develop a multiscale computational approach for heterogeneous Cosserat media without scale separation and high contrast. Cosserat medium has been used in many applications to describe materials whose particles possess rotational degrees of freedom. Many previous findings investigate Cosserat medium in the presence of heterogeneities. Approaches that include homogenization and numerical homogenization have been developed for problems with scale separation, such as periodicity. In these approaches, macroscopic models are derived that show the resulting equations can be Cauchy medium or Cosserat medium depending on the Cosserat intrinsic length scale. In our paper, we consider computational macroscopic models for Cosserat medium without scale separation. We use Generalized Multiscale Finite Element Method (GMsFEM) to derive a macroscopic model on a coarse computational grid, that does not resolve the small scales and the contrast. In this approach, multiple macroscopic basis functions are used to describe the small scales. The resulting macroscopic model is similar to multicontinua model as it contains multiple macroscopic parameters at each macroscopic point (or grid). We present numerical results for different heterogeneous media types. Our numerical results show that using a few multiscale basis functions (i.e., macroscopic parameters), we can achieve a good accuracy. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids. [Copyright &y& Elsevier]

Abstract: We consider numerical algorithms for the simulation of hydrodynamics of two-dimensional vesicles suspended in a viscous Stokesian fluid. The motion of vesicles is governed by the interplay between hydrodynamic and elastic forces. Continuum models of vesicles use a two-phase fluid system with interfacial forces that include tension (to maintain local “surface” inextensibility) and bending. Vesicle flows are challenging to simulate. On the one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives in the bending term. On the other hand, implicit time-stepping schemes can be expensive because they require the solution of a set of nonlinear equations at each time step. Our method is an extension of the work of Veerapaneni et al. [S.K. Veerapaneni, D. Gueyffier, D. Zorin, G. Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, Journal of Computational Physics 228(7) (2009) 2334–2353], in which a semi-implicit time-marching scheme based on a boundary integral formulation of the Stokes problem for vesicles in an unbounded medium was proposed. In this paper, we consider two important generalizations: (i) confined flows within arbitrary-shaped stationary/moving geometries; and (ii) flows in which the interior (to the vesicle) and exterior fluids have different viscosity. In the rest of the paper, we will refer to this as the “viscosity contrast”. These two problems require solving additional integral equations and cause nontrivial modifications to the previous numerical scheme. Our method does not have severe time-step stability constraints and its computational cost-per-time-step is comparable to that of an explicit scheme. The discretization is pseudo-spectral in space, and multistep BDF in time. We conduct numerical experiments to investigate the stability, accuracy and the computational cost of the algorithm. Overall, our method achieves several orders of magnitude speed-up compared to standard explicit schemes. As a preliminary validation of our scheme, we study the dependence of the inclination angle of a single vesicle in shear flow on the viscosity contrast and the reduced area of the vesicle, the lateral migration of vesicles in shear flow, the dispersion of two vesicles, and the effective viscosity of a dilute suspension of vesicles. [Copyright &y& Elsevier]

In this paper, an extension of the DDD-FFT approach presented in [1] is developed for heterogeneous elasticity. For such a purpose, an iterative spectral formulation in which convolutions are calculated in the Fourier space is developed to solve for the mechanical state associated with the discrete eigenstrain-based microstructural representation. With this, the heterogeneous DDD-FFT approach is capable of treating anisotropic and heterogeneous elasticity in a computationally efficient manner. In addition, a GPU implementation is presented to allow for further acceleration. As a first example, the approach is used to investigate the interaction between dislocations and second-phase particles, thereby demonstrating its ability to inherently incorporate image forces arising from elastic inhomogeneities. [ABSTRACT FROM AUTHOR]

In this paper, we consider a high-order discontinuous Galerkin (DG) method for modelling wave propagation in coupled poroelastic–elastic media. The upwind numerical flux is derived as an exact solution for the Riemann problem including the poroelastic–elastic interface. Attenuation mechanisms in both Biot's low- and high-frequency regimes are considered. The current implementation supports non-uniform basis orders which can be used to control the numerical accuracy element by element. In the numerical examples, we study the convergence properties of the proposed DG scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-dimensional nonlinear “multiring” model to compute blood flow in axisymmetric elastic arteries. This model is designed to overcome the numerical difficulties of three-dimensional fluid–structure interaction simulations of blood flow without using the over-simplifications necessary to obtain one-dimensional blood flow models. This multiring model is derived by integrating over concentric rings of fluid the simplified long-wave Navier–Stokes equations coupled to an elastic model of the arterial wall. The resulting system of balance laws provides a unified framework in which both the motion of the fluid and the displacement of the wall are dealt with simultaneously. The mathematical structure of the multiring model allows us to use a finite volume method that guarantees the conservation of mass and the positivity of the numerical solution and can deal with nonlinear flows and large deformations of the arterial wall. We show that the finite volume numerical solution of the multiring model provides at a reasonable computational cost an asymptotically valid description of blood flow velocity profiles and other averaged quantities (wall shear stress, flow rate, ...) in large elastic and quasi-rigid arteries. In particular, we validate the multiring model against well-known solutions such as the Womersley or the Poiseuille solutions as well as against steady boundary layer solutions in quasi-rigid constricted and expanded tubes. [ABSTRACT FROM AUTHOR]

The aim of this article is the construction of a multiphase hyperelastic model. The Eulerian formulation of the hyperelasticity represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. This model is constructed with an elegant approach where the specific energy is given in separable form. The system admits 14 eigenvalues with 7 characteristic eigenfields. The associated Riemann problem is not easy to solve because of the presence of 7 waves. The shear waves are very diffusive when dealing with the full system. In this paper, we use a splitting approach to solve the whole system using 3 sub-systems. This method reduces the diffusion of the shear waves while allowing to use a classical approximate Riemann solver. The multiphase model is obtained by adapting the discrete equations method. This approach involves an additional equation governing the evolution of a phase function relative to the presence of a phase in a cell. The system is integrated over a multiphase volume control. Finally, each phase admits its own equations system composed of three sub-systems. One and three dimensional test cases are presented. [ABSTRACT FROM AUTHOR]

The Dirichlet-to-Neumann finite element method (DtN FEM) has proven to be a powerful numerical approach to solve boundary-value problems formulated in exterior domains. However, its application to elastic semi-infinite domains, which frequently arise in geophysical applications, has been rather limited, mainly due to the lack of explicit closed-form expressions for the DtN map. In this paper, we present a DtN FEM procedure for boundary-value problems of elastostatics in semi-infinite domains with axisymmetry about the vertical axis. A semi-spherical artificial boundary is used to truncate the semi-infinite domain and to obtain a bounded computational domain, where a FEM scheme is employed. By using a semi-analytical procedure of solution in the unbounded residual domain lying outside the artificial boundary, the exact nonlocal boundary conditions provided by the DtN map are numerically approximated and efficiently coupled with the FEM scheme. Numerical results are provided to demonstrate the effectiveness and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the shock transition theory of one-dimensional elastic-perfectly plastic solid. The emphasis focuses on the properties of the elastic Hugoniot curves in Wilkins model with general Grüneisen-type equation of state (EOS) and arbitrary state ahead of a shock. These works can be generalized to plastic case. [ABSTRACT FROM AUTHOR]

We develop a fourth order accurate finite difference method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson (2012) [11] . The proposed method discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite difference method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. We also generalize and evaluate the super-grid far-field technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. As a result, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that sufficiently wide super-grid layers result in very small artificial reflections. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals. [ABSTRACT FROM AUTHOR]

In this paper we describe a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combined-field boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developed by Ganesh and Hawkins [20] – for solving second kind boundary integral equations in electromagnetism – to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parametrised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three-dimensional convex and non-convex smooth obstacles. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of the three-dimensional fluid-structure interaction problem in haemodynamics, in the case of real geometries, physiological data and finite elasticity vessel deformations. We study some new inexact schemes, obtained from semi-implicit approximations, which treat exactly the physical interface conditions while performing just one or few iterations for the management of the interface position and of the fluid and structure non-linearities. We show that such schemes allow to improve the efficiency while preserving the accuracy of the related exact (implicit) scheme. To do this we consider both a simple analytical test case and two real cases of clinical interest in haemodynamics. We also provide an error analysis for a simple differential model problem when a BDF method is considered for the time discretization and only few Newton iterations are performed at each temporal instant. [ABSTRACT FROM AUTHOR]

Abstract: A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations. In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simple criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed. [Copyright &y& Elsevier]

Abstract: A tensile wave can be destructive, as it can generate microcracks and cause the interface detachments in a composite structure. Recently Huang, Dai, Chen and Kong [14] proposed to use material nonlinearity to generate wave catching-up phenomena such that a tensile wave can catch the first transmitted compressive wave, leading to the reduction of the former. They gave the existence and asymptotic solutions in short time. In this paper, we develop a dissipation-rate reserving discontinuous Galerkin (DG) method to solve numerically the system of balance laws with discontinuous flux and inflow boundary condition for waves in a two-material nonlinearly elastic composite bar. Physically, the aim is to examine the reduction ratio of the tensile wave due to wave catching-up phenomena in long time. The stability analysis for the DG method is also presented. And, a contribution is that the numerical fluxes are chosen to satisfy the local dissipation rate across a shock. Numerical results are compared with the available asymptotic solutions in short time and good agreements are found. In particular, the numerical solutions give a high resolution for various interactions between the tensile shock, compressive shock and rarefaction wave. The results confirm that when the stress-strain relation of the second nonlinearly elastic material is convex, indeed the tensile wave can catch the previously transmitted compressive wave. The simulation results by the DG method show that the catching-up phenomena can reduce the magnitude of the tensile wave by more than 400%. We also examine how material parameters influence the reduction ratio. As a verification, the solutions by a weighted essentially non-oscillatory (WENO) finite volume method are also presented, which are in agreement with those by the DG method. [Copyright &y& Elsevier]

Abstract: This paper presents our new solver BCCG+FAI for solving elastic normal contact problems. This is a comprehensible approach that is based on the Conjugate Gradients (CG) algorithm and that uses FFTs. A first novel aspect is the definition of the “FFT-based Approximate Inverse” preconditioner. The underlying idea is that the inverse matrix can be approximated well using a Toeplitz or block-Toeplitz form, which can be computed using the FFT of the original matrix elements. This preconditioner makes the total number of CG iterations effectively constant in 2D and very slowly increasing in 3D problems. A second novelty is how we deal with a prescribed total force. This uses a deflation technique in such a way that CGs convergence and finite termination properties are maintained. Numerical results show that this solver is more effective than existing CG-based strategies, such that it can compete with Multi-Grid strategies over a much larger problem range. In our opinion it could be the new method of choice because of its simple structure and elegant theory, and because robust performance is achieved independently of any problem specific parameters. [Copyright &y& Elsevier]

Abstract: In this paper, we describe a cell-centered Lagrangian scheme devoted to the numerical simulation of solid dynamics on two-dimensional unstructured grids in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic–plastic flow, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the deviatoric stress, its time evolution is governed by a classical constitutive law for isotropic material. The plasticity model employs the von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical scheme relies on a finite volume cell-centered method wherein numerical fluxes are expressed in terms of sub-cell force. The generic form of the sub-cell force is obtained by requiring the scheme to satisfy a semi-discrete dissipation inequality. Sub-cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node-centered solver, which results from total energy conservation. The nominally second-order extension is achieved by developing a two-dimensional extension in the Lagrangian framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Falcovitz [M. Ben-Artzi, J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monogr. Appl. Comput. Math. (2003)]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases. [Copyright &y& Elsevier]

Abstract: We investigate the curvature effect of a thin, curved elastic interface that separates two subdomains and exerts a pressure due to a curvature effect. This pressure, which we refer to as interface pressure, is similar to the surface tension in fluid mechanics. It is important in some applications, such as the canopy of parachutes, biological membranes of cells, balloons, airbags, etc., as it partially balances a pressure jump between the two sides of an interface. In this paper, we show that the interface pressure is equal to the trace of the matrix product of the curvature tensor and the Cauchy stress tensor in the tangent plane. We derive the theory for interfaces in both 2-D and 3-D, and present numerical discretizations for computing the quality over triangulated surfaces. [Copyright &y& Elsevier]

Abstract: In this paper exponential basis functions (EBFs) satisfying the governing equations of elastic problems with incompressible materials are introduced. Due to similarity between elasticity problems and steady state fluid problems the bases found for the former problems are used for latter problems. We discuss on using single field form known as displacement/velocity based formulation and also on using a two-field form known as u -p formulation. In the first formulation we find the pressure bases through performing a limit analysis using a fictitious bulk modulus while in the second formulation the bases are found directly by considering the pressure as a separate variable. In the second formulation we directly apply the condition of incompressibility. It is shown that both formulations yield identical bases meaning that the first one may be used in a standard approach. However, it is also shown that when the incompressibility condition is applied by a Laplacian of pressure in the second formulation, some additional spurious EBFs may be obtained. Having defined appropriate bases, we follow the solution strategy recently introduced by the authors for other engineering problems. Some well-known benchmark problems are solved to show the capabilities of the method. [Copyright &y& Elsevier]

Abstract: We analyze a new mathematical and numerical framework, the “Voronoi Implicit Interface Method” (“VIIM”), first introduced in Saye and Sethian (2011) [R.I. Saye, J.A. Sethian, The Voronoi Implicit Interface Method for computing multiphase physics, PNAS 108 (49) (2011) 19498–19503] for tracking multiple interacting and evolving regions (“phases”) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.). From a mathematical point of view, the method provides a theoretical framework for moving interface problems that involve multiple junctions, defining the motion as the formal limit of a sequence of related problems. Discretizing this theoretical framework provides a numerical methodolology which automatically handles multiple junctions, triple points and quadruple points in two dimensions, as well as triple lines, etc. in higher dimensions. Topological changes in the system occur naturally, with no surgery required. In this paper, we present the method in detail, and demonstrate several new extensions of the method to different physical phenomena, including curvature flow with surface energy densities defined on a per-phase basis, as well as multiphase fluid flow in which density, viscosity and surface tension can be defined on a per-phase basis. We test this method in a variety of ways. We perform rigorous analysis and demonstrate convergence in both two and three dimensions for a variety of evolving interface problems, including verification of von Neumann–Mullins’ law in two dimensions (and its analog in three dimensions), as well as normal driven flow and curvature flow with and without constraints, demonstrating topological change and the effects of different boundary conditions. We couple the method to a second order projection method solver for incompressible fluid flow, and study the effects of membrane permeability and impermeability, large shearing torsional forces, and the effects of varying density, viscosity and surface tension on a per-phase basis. Finally, we demonstrate convergence in both space and time of a topological change in a multiphase foam. [Copyright &y& Elsevier]

Abstract: This paper is devoted to developing a multi-material numerical scheme for non-linear elastic solids, with emphasis on the inclusion of interfacial boundary conditions. In particular for colliding solid objects it is desirable to allow large deformations and relative slide, whilst employing fixed grids and maintaining sharp interfaces. Existing schemes utilising interface tracking methods such as volume-of-fluid typically introduce erroneous transport of tangential momentum across material boundaries. Aside from combatting these difficulties one can also make improvements in a numerical scheme for multiple compressible solids by utilising governing models that facilitate application of high-order shock capturing methods developed for hydrodynamics. A numerical scheme that simultaneously allows for sliding boundaries and utilises such high-order shock capturing methods has not yet been demonstrated. A scheme is proposed here that directly addresses these challenges by extending a ghost cell method for gas-dynamics to solid mechanics, by using a first-order model for elastic materials in conservative form. Interface interactions are captured using the solution of a multi-material Riemann problem which is derived in detail. Several different boundary conditions are considered including solid/solid and solid/vacuum contact problems. Interfaces are tracked using level-set functions. The underlying single material numerical method includes a characteristic based Riemann solver and high-order WENO reconstruction. Numerical solutions of example multi-material problems are provided in comparison to exact solutions for the one-dimensional augmented system, and for a two-dimensional friction experiment. [Copyright &y& Elsevier]

Abstract: In this paper, a phase field model is developed for vesicle adhesion involving complex substrate and vesicle geometries. The model takes into account an adhesion potential that depends on the distance of vesicle to the substrate. A variational problem is solved in a 3D computational domain by minimizing the contribution of bending elastic energy and the adhesion energy under the constraints of total surface area and volume, described via a phase function. An adaptive finite element method is used to efficiently compute the numerical solutions of the model. The computational results are validated through comparison of several axisymmetric shapes with the sharp-interface ODE solution. Moreover, we compute shapes for non-axisymmetric situations to support the observation that concave substrates favor adhesion. [Copyright &y& Elsevier]

Abstract: Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented. [Copyright &y& Elsevier]

Abstract: In this paper, the dynamics of two dimensional elastic particles in a Newtonian viscous shear flow is studied numerically. To describe the elastic deformation, an evolution equation for the Eulerian Almansi strain tensor is derived. A constitutive equation is thus constructed for an incompressible “Neo–Hookean” elastic solid where the extra stress tensor is assumed to be linearly proportional to the Almansi strain tensor. The displacement field does not appear in this formulation. A monolithic finite element solver which uses Arbitrary Lagrangian–Eulerian moving mesh technique is then implemented to solve the velocity, pressure and stress in both fluid and solid phase simultaneously. It is found that the deformation of the particle in the shear flow is governed by two non-dimensional parameters: Reynolds number (Re) and Capillary number (Ca, which is defined as the ratio of the viscous force to the elastic force). In the Stokes flow regime and when Ca is small (), the particle deforms into an elliptic shape while the material points inside the particle experience a tank-treading like motion with a steady velocity field. The deformation of the elastic particle is observed to vary linearly with Ca, which agrees with theoretical results from a perturbation analysis. Interactions between two particles in a viscous shear flow are also explored. It is observed that after the initial complicated interactions, both particles reach an equilibrium elliptic shape which is consistent with that of a single particle. [Copyright &y& Elsevier]

The aim of this paper is to propose a new method to solve the inverse scattering problem in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine the Young modulus of elastic unknown "inclusions", i.e. not observable solid objects, located in the elastic medium, from partial aperture boundary measurements. This approach combines a time-reversal approach and redatuming - a technique that consists in virtually moving the sensors from the original acquisition location to an arbitrary position - which allows to reduce the size of the inspected medium. Thanks to this strategy, we can recover the properties of the scatterer, this technique being also relatively insensitive to noise in the data. • Acoustic recorded data contain enough information to identify elastic inclusions. • Reducing the size of the computational domain increases the success rate. • Using adapted redatuming helps to reduce the size of the domain and to determine the elastic scatterers properties. • This approach is fairly insensitive to noise in the data. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss a simple iterative-perturbation scheme for solving the elasticity equation in systems with strong elastic inhomogeneity. As an example, a thin film in contact with a gas and a substrate is considered. The scheme is demonstrated to be efficient through numerical experiments and reliable through rigorous mathematical justification. It is then applied to the study of the inhomogeneous shear modulus effect on the microstructure evolution in thin films based on the phase-field method. [Copyright &y& Elsevier]

In this paper we construct numerical schemes of high order of accuracy for hyperbolic balance law systems with spatially variable flux function and a source term of the geometrical type. We start with the original finite difference characteristicwise weighted essentially nonoscillatory (WENO) schemes and then we create new schemes by modifying the flux formulations (locally Lax-Friedrichs and Roe with entropy fix) in order to account for the spatially variable flux, and by decomposing the source term in order to obtain balance between numerical approximations of the flux gradient and of the source term. We apply so extended WENO schemes to the one-dimensional open channel flow equations and to the one-dimensional elastic wave equations. In particular, we prove that in these applications the new schemes are exactly consistent with steady-state solutions from an appropriately chosen subset. Experimentally obtained orders of accuracy of the extended and original WENO schemes are almost identical on a convergence test. Other presented test problems illustrate the improvement of the proposed schemes relative to the original WENO schemes combined with the pointwise source term evaluation. As expected, the increase in the formal order of accuracy of applied WENO reconstructions in all the tests causes visible increase in the high resolution properties of the schemes. [Copyright &y& Elsevier]

In this paper, we compute the equilibrium configurations of a vesicle membrane under elastic bending energy, with prescribed volume and surface area. A variational phase field method is developed for such a problem. Discrete finite difference approximations and numerical simulations are carried out in the axial symmetrical cases. Different energetic bifurcation phenomena are discussed. [Copyright &y& Elsevier]

Mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the Eulerian fluid domain is not known a priori – it is at least partially determined by the displacement of the elastic solid. In this paper, a first-order system least squares finite element formulation is used to solve the nonlinear system of model equations using different iteration techniques, including an approach where the equations are fully coupled and two other approaches in which the equations are decoupled. The discrete linear system of equations is solved using an algebraic multigrid solver as a preconditioner for a conjugate gradient iteration. The numerical results show that the approach is optimal in the sense that computational cost is proportional to the degrees of freedom. The results also show that the choice of iteration method, from fully coupled to fully decoupled, does not significantly effect computational cost, but it does influence the error in the solution. [Copyright &y& Elsevier]

We present a numerical method to take into account 2D arbitrary-shaped interfaces in classical finite-difference schemes, on a uniform Cartesian grid. This work extends the “explicit simplified interface method” (ESIM), previously proposed in 1D [J. Comput. Phys. 168 (2001) 227–248]. The physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics, with stationary interfaces. Our method maintains, near the interfaces, properties of the schemes in homogeneous medium, such as the order of accuracy and the stability limit. Moreover, it enforces the numerical solution to satisfy the exact interface conditions. Lastly, it provides subcell geometrical features of the interface inside the meshing. The ESIM can be coupled automatically with a wide class of numerical schemes (Lax–Wendroff, flux-limiter schemes, etc.) for a negligible additional computational cost. Throughout the paper, we focus on the challenging case of an interface between a fluid and an elastic solid. In numerical experiments, we provide comparisons between numerical solutions and original analytic solutions, showing the efficiency of the method. [Copyright &y& Elsevier]

In this paper, we present a general iterative method for the solution of the Riemann problem for hyperbolic systems of PDEs. The method is based on the multiple shooting method for free boundary value problems. We demonstrate the method by solving one-dimensional Riemann problems for hyperelastic solid mechanics. Even for conditions representative of routine laboratory conditions and military ballistics, dramatic differences are seen between the exact and approximate Riemann solution. The greatest discrepancy arises from misallocation of energy between compressional and thermal modes by the approximate solver, resulting in nonphysical entropy and temperature estimates. Several pathological conditions arise in common practice and modifications to the method to handle these are discussed. These include points where genuine nonlinearity is lost, degeneracies, and eigenvector deficiencies that occur upon melting. [Copyright &y& Elsevier]

• First numerical method to incorporate the self consistent Ball-Majumdar potential. • Method guarantees the eigenvalues of the nematic Q-tensor remain physically realistic. • Mean field method allows anisotropic elasticity in a Q-tensor representation. We present a numerical method, based on a tensor order parameter description of a nematic phase, that allows fully anisotropic elasticity. Our method thus extends the Landau-de Gennes Q -tensor theory to anisotropic phases. A microscopic model of the nematogen is introduced (the Maier-Saupe potential in the case discussed in this paper), combined with a constraint on eigenvalue bounds of Q. This ensures a physically valid order parameter Q (i.e., the eigenvalue bounds are maintained), while allowing for more general gradient energy densities that can include cubic nonlinearities, and therefore elastic anisotropy. We demonstrate the method in two specific two dimensional examples in which the Landau-de Gennes model including elastic anisotropy is known to fail, as well as in three dimensions for the cases of a hedgehog point defect, a disclination line, and a disclination ring. The details of the numerical implementation are also discussed. [ABSTRACT FROM AUTHOR]

• We construct the surface equivalent of the upper-convected Maxwell model. • The model is used to build a numerical method of a viscoelastic surface fluid. • Numerical results indicate good agreement with analytical solutions. • First simulations of a viscoelastic fluid surface in shear flow are provided. Viscoelastic surface rheology plays an important role in multiphase systems. A typical example is the actin cortex which surrounds most animal cells. It shows elastic properties for short time scales and behaves viscous for longer time scales. Hence, realistic simulations of cell shape dynamics require a model capturing the entire elastic to viscous spectrum. However, currently there are no numerical methods to simulate deforming viscoelastic surfaces. Thus models for the cell cortex, or other viscoelastic surfaces, are usually based on assumptions or simplifications which limit their applicability. In this paper we develop a first numerical approach for simulation of deforming viscoelastic surfaces. To this end, we derive the surface equivalent of the upper convected Maxwell model using the GENERIC formulation of nonequilibrium thermodynamics. The model distinguishes between shear dynamics and dilatational surface dynamics. The viscoelastic surface is embedded in a viscous fluid modelled by the Navier-Stokes equation. Both systems are solved using Finite Elements. The fluid and surface are combined using an Arbitrary Lagrange-Eulerian (ALE) Method that conserves the surface grid spacing during rotations and translations of the surface. We verify this numerical implementation against analytic solutions and find good agreement. To demonstrate its potential we simulate the experimentally observed tumbling and tank-treading of vesicles in shear flow. We also supply a phase-diagram to demonstrate the influence of the viscoelastic parameters on the behaviour of a vesicle in shear flow. Finally, we explore cytokinesis as a future application of the numerical method by simulating the start of cytokinesis using a spatially dependent function for the surface tension. [ABSTRACT FROM AUTHOR]

• The data, recorded in the acoustic part of the medium (basically scalar quantities) contain enough information to identify the properties and locations of the inclusions located in the elastic part. This was made possible by introducing criteria allowing us construct images of these inclusions. • Using the same procedure, one can differentiate between benign and malignant inclusions. • This approach is fairly insensitive to noise in the data. The aim of this paper is to study the feasibility of time-reversal methods in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine, from partial aperture boundary measurements, the presence and some physical properties of elastic unknown "inclusions", i.e. not observable solid objects, located in the elastic medium. We first derive a variational formulation of the acousto-elastic problem, from which one constructs a time-dependent finite element method to solve the forward, and then, the time reversed problem. Several criteria, derived from the reverse time migration framework, are then proposed to construct images of the inclusions, and to determine their locations. The dependence of the approach to several parameters (aperture, number of sources, etc.) is also investigated. In particular, it is shown that one can differentiate between a benign and malignant close inclusions. This technique is fairly insensitive to noise in the data. [ABSTRACT FROM AUTHOR]

In this paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to first solving a nonlinear poroelasticity problem. The arising system consists of a nonlinear pressure equation and a nonlinear stress equation in strain-limiting setting, where strains keep bounded while stresses can grow arbitrarily large. After time-discretization of the system, to tackle the nonlinearity, we linearize the resulting equations by Picard iteration. To handle the linearized equations, we employ the CEM-GMsFEM and obtain appropriate offline multiscale basis functions for the pressure and the displacement. More specifically, first, auxiliary multiscale basis functions are generated by solving local spectral problems, via the GMsFEM. Then, multiscale spaces are constructed in oversampled regions, by solving a constraint energy minimizing (CEM) problem. After that, this strategy (with the CEM-GMsFEM) is also applied to a static case of the above nonlinear poroelasticity problem, that is, elasticity problem, where the residual based online multiscale basis functions are generated by an adaptive enrichment procedure, to further reduce the error. Convergence of the two cases is demonstrated by several numerical simulations, which give accurate solutions, with converging coarse-mesh sizes as well as few basis functions (degrees of freedom) and oversampling layers. [ABSTRACT FROM AUTHOR]

The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. Since there is no explicit boundary representation in density-based topology optimization, the design-dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field. The density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of PDE. The Heaviside projection, density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it. For problems with both loading and non-loading boundary, an auxiliary density field is introduced to keep track of the loading boundary. Both 2D and 3D heat conduction problems, linear elasticity problems and coupled thermoelastic problems under design-dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method. [ABSTRACT FROM AUTHOR]

• A meshfree LRBFCM based on the nonlocal elasticity theory is developed for wave propagation analysis in nanoscale phononic crystals. • The band structures of 1D nanoscale phononic crystals obtained by the LRBFCM agree well with the reference results. • The band structures and transmission coefficients of elastic waves in 2D nanoscale phononic crystals are computed and analyzed by the LRBFCM. In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures. [ABSTRACT FROM AUTHOR]

We study the interior transmission eigenvalue problem for the elastic wave scattering in this paper. We aim to show the distribution of positive eigenvalues by efficient numerical algorithms. Here the elastic waves are scattered by the perturbations of medium parameters, which include the elasticity tensor C and the density ρ. Let us denote (C 0 , ρ 0) and (C 1 , ρ 1) the background and the perturbed medium parameters, respectively. We consider two cases of perturbations, C 0 = C 1 , ρ 1 ≠ ρ 0 (case 1) and C 0 ≠ C 1 , ρ 1 = ρ 0 (case 2). After discretizing the associated PDEs by FEM, we are facing the computation of generalized eigenvalues problems (GEP) with matrices of large size. These GEPs contain huge number of nonphysical zeros (for case 1) or nonphysical infinities (for case 2). In order to locate several hundred positive eigenvalues effectively, we then convert GEPs to suitable quadratic eigenvalues problems (QEP). We then implement a quadratic Jacobi-Davidson method combining with partial locking or partial deflation techniques to compute 500 positive eigenvalues. [ABSTRACT FROM AUTHOR]