Discontinuous Material Properties Research Articles

Finite volume Ghost Fluid Method implementation of interfacial forces in PISO loop

In this work, we implement interfacial forces (surface tension, hydrostatic and viscous forces) by enlisting the finite volume discretization of GFM (Ghost Fluid Method) using A-CLSVOF (Algebraic Coupled Level-Set/Volume-of-Fluid) method for the mass conservative, and smooth interface description. The widely used PISO momentum solution to resolve the pressure–velocity coupling is presented along with the present GFM discretization and its placement within the PISO loop. The pressure jump at the interface due to the interfacial forces is made sharp via direct calculation of the modified pressure matrix coefficients corresponding to targeted interfacial cells, and as a source term for the jump value itself. The Level-Set field is enlisted for curvature computation in A-CLSVOF and for the interpolation and weighting of the relative contribution of the capillary force in adjacent for the matrix coefficients in the FV framework. To assess the A-CLSVOF/GFM performance, four canonical cases were studied. In the case of a static droplet in suspension, A-CLSVOF/GFM produces a sharp and accurate pressure jump compared to the traditional CSF implementation of A-CLSVOF. The interaction of viscous and capillary forces is proven to be accurate and consistent with theoretical results for the classical capillary wave. For the linear two-layer shear flow, GFM sharp treatment of the viscosity captured the velocity gradient across the interface and removed the diffusion of the viscous stresses caused by the discontinuous material properties. Finally, the combination of all GFM improvements proposed in this study are compared to experimental findings of terminal velocity for a gaseous bubble rising in a viscous fluid. GFM outperforms CSF with errors of 4.6% and 14.0% respectively.

High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) methods using ultraweak formulations

This work represents the first endeavor in using ultraweak formulations to implement high-order polygonal finite element methods via the discontinuous Petrov–Galerkin (DPG) methodology. Ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space, while most of the trial space can be chosen as copies of L2-discretizations that have no need to be continuous across adjacent elements. Additionally, the test spaces are broken along the mesh interfaces. This allows one to construct conforming polygonal finite element methods, termed here as PolyDPG methods, by defining most spaces by restriction of a bounding triangle or box to the polygonal element. The only variables that require nontrivial compatibility across elements are the so-called interface or skeleton variables, which can be defined directly on the element boundaries. Unlike other high-order polygonal methods, PolyDPG methods do not require ad hoc stabilization terms thanks to the crafted stability of the DPG methodology. A proof of convergence of the form hp is provided and corroborated through several illustrative numerical examples. These include polygonal meshes with n-sided convex elements and with highly distorted concave elements, as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid. Since PolyDPG methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes. This work is also accompanied by an open-source PolyDPG software supporting polygonal and conventional elements.

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.

Improved PISO algorithms for modeling density varying flow in conjugate fluid–porous domains

Two modified segregated PISO algorithms are proposed, which are constructed to avoid the development of spurious oscillations in the computed flow near sharp interfaces of conjugate fluid-porous domains. The new collocated finite volume algorithms modify the Rhie-Chow interpolation to maintain a correct pressure-velocity coupling when large discontinuous momentum sources associated with jumps in the local permeability and porosity are present. The Re-Distributed Resistivity (RDR) algorithm is based on spreading flow resistivity over the grid cells neighboring a discontinuity in material properties of the porous medium. The Face Consistent Pressure (FCP) approach derives an auxiliary pressure value at the fluid-porous interface using momentum balance around the interface. Such derived pressure correction is designed to avoid spurious oscillations as would otherwise arise with a strictly central discretization. The proposed algorithms are successfully compared against published data for the velocity and pressure for two reference cases of viscous flow. The robustness of the proposed algorithms is additionally demonstrated for strongly reduced viscosity, i.e., higher Reynolds number flows and low Darcy numbers, i.e., low permeability of the porous regions in the domain, for which solutions without unphysical oscillations are computed. Both RDR and FCP are found to accurately represent porous media flow near discontinuities in material properties on structured grids. Two modified segregated PISO algorithms are proposed.The Rhie-Chow interpolation is modified to maintain a correct pressure-velocity coupling when large discontinuous momentum sources are present.The proposed algorithms are successfully compared against published data for the velocity and pressure for two reference cases of viscous flow.The robustness of the proposed algorithms is additionally demonstrated for high Reynolds number flows and low Darcy numbers.

Interface- and discontinuity-aware numerical schemes for plasma 3-T radiation diffusion in two and three dimensions

A set of numerical schemes is developed for two- and three-dimensional time-dependent 3-T radiation diffusion equations in systems involving multi-materials. To resolve sub-cell structure, interface reconstruction is implemented within any cell that has more than one material. Therefore, the system of 3-T radiation diffusion equations is solved on two- and three-dimensional polyhedral meshes. The focus of the development is on the fully coupling between radiation and material, the treatment of nonlinearity in the equations, i.e., in the diffusion terms and source terms, treatment of the discontinuity across cell interfaces in material properties, the formulations for both transient and steady states, the property for large time steps, and second order accuracy in both space and time. The discontinuity of material properties between different materials is correctly treated based on the governing physics principle for general polyhedral meshes and full nonlinearity. The treatment is exact for arbitrarily strong discontinuity. The scheme is fully nonlinear for the full nonlinearity in the 3-T diffusion equations. Three temperatures are fully coupled and are updated simultaneously. The scheme is general in two and three dimensions on general polyhedral meshes. The features of the scheme are demonstrated through numerical examples for transient problems and steady states. The effects of some simplifications of numerical schemes are also shown through numerical examples, such as linearization, simple average of diffusion coefficient, and approximate treatment for the coupling between radiation and material.

Variational time discretization for mixed finite element approximations of nonstationary diffusion problems

We develop and study numerically two families of variational time discretization schemes for mixed finite element approximations applied to nonstationary diffusion problems. Continuous and discontinuous approximations of the time variable are encountered. The solution of the arising algebraic block system of equations by a Schur complement technique is described and an efficient preconditioner for the iterative solution process is constructed. The expected higher order rates of convergence are demonstrated in numerical experiments. Moreover, superconvergence properties are verified. Further, the efficiency and stability of the approaches are illustrated for a more sophisticated three-dimensional application of practical interest with discontinuous and anisotropic material properties.

Finite strain primal interface formulation with consistently evolving stabilization

SummaryA stabilized discontinuous Galerkin method is developed for general hyperelastic materials at finite strains. Starting from a mixed method incorporating Lagrange multipliers along the interface, the displacement formulation is systematically derived through a variational multiscale approach whereby the numerical fine scales are modeled via edge bubble functions. Analytical expressions that are free from user‐defined parameters arise for the weighted numerical flux and stability tensor. In particular, the specific form taken by these derived quantities naturally accounts for evolving geometric nonlinearity as well as discontinuous material properties. The method is applicable both to problems containing nonconforming meshes or different element types at specific interfaces and to problems consisting of fully discontinuous numerical approximations. Representative numerical tests involving large strains and rotations are performed to confirm the robustness of the method. Copyright © 2015 John Wiley & Sons, Ltd.

Second-order accurate interface- and discontinuity-aware diffusion solvers in two and three dimensions

A numerical scheme is developed for two- and three-dimensional time-dependent diffusion equations in numerical simulations involving mixed cells. The focus of the development is on the formulations for both transient and steady states, the property for large time steps, second-order accuracy in both space and time, the correct treatment of the discontinuity in material properties, and the handling of mixed cells. For a mixed cell, interfaces between materials are reconstructed within the cell so that each of resulting sub-cells contains only one material and the material properties of each sub-cell are known. Diffusion equations are solved on the resulting polyhedral mesh even if the original mesh is structured. The discontinuity of material properties between different materials is correctly treated based on governing physics principles. The treatment is exact for arbitrarily strong discontinuity. The formulae for effective diffusion coefficients across interfaces between materials are derived for general polyhedral meshes. The scheme is general in two and three dimensions. Since the scheme to be developed in this paper is intended for multi-physics code with adaptive mesh refinement (AMR), we present the scheme on mesh generated from AMR. The correctness and features of the scheme are demonstrated for transient problems and steady states in one-, two-, and three-dimensional simulations for heat conduction and radiation heat transfer. The test problems involve dramatically different materials.

A second order virtual node algorithm for Navier–Stokes flow problems with interfacial forces and discontinuous material properties

We present a numerical method for the solution of the Navier–Stokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and velocity boundary conditions on irregular domains. The method improves on results in [1] (which solved the Stokes equations in two dimensions) by providing treatment of fluid inertia as well as a new discretization of jump and boundary conditions that accurately resolves null modes in both two and three dimensions. We discretize the equations using an embedded approach on a uniform MAC grid to yield discretely divergence-free velocities that are second order accurate. We maintain our interface using the level set method or, when more appropriate, the particle level set method. We show how to implement Dirichlet (known velocity), Neumann (known normal stress), and slip velocity boundary conditions as special cases of our interface representation. The method leads to a discrete, symmetric KKT system for velocities, pressures, and Lagrange multipliers. We also present a novel simplification to the standard combination of the second order semi-Lagrangian and BDF schemes for discretizing the inertial terms. Numerical results indicate second order spatial accuracy for the velocities (L∞ and L2) and first order for the pressure (in L∞, second order in L2). Our temporal discretization is also second order accurate.

P1/P0+ elements for incompressible flows with discontinuous material properties

We present a 3-noded triangle and a 4-noded tetrahedra with a continuous linear velocity and a discontinuous linear pressure field formed by the sum of an unknown constant pressure field and a prescribed linear field that satisfies the steady state momentum equations for a constant body force. The elements are termed P1/P0+ as the “effective” pressure field is linear, although the unknown pressure field is piecewise constant within each element. The elements have an excellent behavior for incompressible viscous flow problems with discontinuous material properties formulated in either Eulerian or Lagrangian descriptions. The necessary numerical stabilization for dealing with the inf–sup condition imposed by the incompressibility constraint and high convective effects (in Eulerian flows) is introduced via the Finite Calculus (FIC) approach. For the sake of clarity, the element derivation is presented first for the simpler Stokes equations written in the standard Eulerian frame. The extension of the formulation to the Navier–Stokes equations written in the Eulerian and Lagrangian frameworks is straightforward and is presented in the second part of the paper.The efficiency and accuracy of the new P1/P0+ triangle is verified by solving a set of incompressible multifluid flow problems using a Lagrangian approach and a classical Eulerian description. The excellent performance of the new triangular element in terms of mass conservation and general accuracy for analysis of fluids with discontinuous material properties is highlighted.

Stress analysis of multi-layered hollow anisotropic composite cylindrical structures using the homogenization method

This paper presents a general and efficient stress analysis strategy for hollow composite cylindrical structures consisting of multiple layers of different anisotropic materials subjected to different loads. Cylindrical material anisotropy and various loading conditions are considered in the stress analysis. The general stress solutions for homogenized hollow anisotropic cylinders subjected to pressure, axial force, torsion, shear and bending are presented with explicit formulations under typical force and displacement boundary conditions. The stresses and strains in a layer of the composite cylindrical structures are obtained from the solutions of homogenized hollow cylinders with effective material properties and discontinuous layer material properties. Effective axial, torsional, bending and coupling stiffness coefficients taking into account material anisotropy are also determined from the strain solutions for the hollow composite cylindrical structures. Examples show that the material anisotropy may have significant effects on the effective stiffness coefficients in some cases. The stress analysis method is demonstrated with an example of stress analysis of a 22-layer composite riser, and the results are compared with numerical solutions. This method is efficient for stress analysis of thin-walled or moderately thick-walled hollow composite cylindrical structures with various multiple layers of different materials or arbitrary fiber angles because no explicit interfacial continuity parameters are required. It provides an efficient and easy-to-use analysis tool for assessing hollow composite cylindrical structures in engineering applications.

Material property discontinuities in intervertebral disc porohyperelastic finite element models generate numerical instabilities due to volumetric strain variations

Numerical studies of the intervertebral disc (IVD) are important to better understand the load transfer and the mechanobiological processes within the disc. Among the relevant calculations, fluid-related outputs are critical to describe and explore accurately the tissue properties. Porohyperelastic finite element models of IVD can describe accurately the disc behaviour at the organ level and allow the inclusion of fluid effects. However, results may be affected by numerical instabilities when fast load rates are applied. We hypothesized that such instabilities would appear preferentially at material discontinuities such as the annulus–nucleus boundary and should be considered when testing mesh convergence. A L4–L5 IVD model including the nucleus, annulus and cartilage endplates were tested under pure rotational loads, with different levels of mesh refinement. The effect of load relaxation and swelling were also studied. Simulations indicated that fluid velocity oscillations appeared due to numerical instability of the pore pressure spatial derivative at material discontinuities. Applying local refinement only was not enough to eliminate these oscillations. In fact, mesh refinements had to be local, material-dependent, and supplemented by the creation of a material transition zone, including interpolated material properties. Results also indicated that oscillations vanished along load relaxation, and faster attenuation occurred with the incorporation of the osmotic pressure. We concluded that material discontinuities are a major cause of instability for poromechanical calculations in multi-tissue models when load velocities are simulated. A strategy was presented to address these instabilities and recommendations on the use of IVD porohyperelastic models were given.

A second order virtual node algorithm for Stokes flow problems with interfacial forces, discontinuous material properties and irregular domains

We present numerical methods for the solution of the Stokes equations that handle interfacial discontinuities, discontinuous material properties and irregular domains. The discretization couples a Lagrangian representation of the material interface with Eulerian representations of the fluid velocity and pressure. The methods are efficient, easy to implement and yield discretely divergence-free velocities that are second order accurate in L∞. For the special case of continuous fluid viscosity, we present a method that decouples the Stokes equations into three Poisson interface problems which we use the techniques in Bedrossian (2010) [1] to solve. We also solve a fourth Poisson equation to enforce a discrete divergence free condition in this case. We discretize all equations using an embedded approach on a uniform MAC grid employing virtual nodes and duplicated cells at the interfaces. These additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface. In the case of discontinuous viscosity, we require a Lagrange multiplier term to enforce continuity of the fluid velocity. We provide a novel discretization of this term that accurately resolves constant pressure null modes. We show that the accurate resolution of these modes significantly improves performance. The discrete coupled equations for the velocity, pressure and Lagrange multipliers are in the form of a symmetric KKT system. However, if both fluids have the same viscosity then all four linear systems involved are symmetric positive definite with three of the four having the standard 5-point Laplace stencil everywhere. Numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure in the general case. For the continuous viscosity case, numerical results indicate second order accuracy for both velocities and pressure.

Defining regions of interest for microwave imaging using near-field reflection data

Microwave imaging benefits from information on the internal structure of the object of interest. Previously, we developed a tool to define regions dominated by a particular material and tested this approach with 2-D simulations of breast models. In this paper, we apply the technique to experimental data with the goal to extract an object's internal structural information using radar-based techniques. Specifically, the technique extracts information from microwave reflection data in order to identify discontinuities in material properties. By combining observations from multiple antenna locations, this information is used to estimate regions of interest. The utility of the tool is demonstrated in practical scenarios where the data are generated experimentally from cylindrical models using an ultra-wideband sensor.

Simulation of void formation in liquid composite molding processes

Air entrapment within and between fiber tows during preform permeation in liquid composite molding (LCM) processes leads to undesirable quality in the resulting composite material with defects such as discontinuous material properties, failure zones, and visual flaws. Essential to designing processing conditions for void-free filling is the development of an accurate prediction of local air entrapment locations as the resin permeates the preform. To this end, the study presents a numerical simulation of the infiltrating dual-scale resin flow through the actual architecture of plain weave fibrous preforms accounting for the capillary effects within the fiber bundles. The numerical simulations consider two-dimensional cross sections and full three-dimensional representations of the preform to investigate the relative size and location of entrapped voids for a wide range of flow, preform geometry, and resin material properties. Based on the studies, a generalized paradigm is presented for predicting the void content as a function of the Capillary and Reynolds numbers governing the materials and processing. Optimum conditions for minimizing air entrapment during processing are also presented and discussed.

Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions

AbstractWe formulate a one-dimensional time-dependent non-linear mathematical model for some types of physiological fluid flow in collapsible tubes with discontinuous material properties. The resulting 6 × 6 hyperbolic system is analysed and the associated Riemann problem is solved exactly. Although the solution algorithm deals with idealised cases, it is nonetheless uniquely well-suited for assessing the performance of numerical methods intended for simulating more general situations. Moreover, our model may be a useful starting point for numerical calculations of realistic flows involving rapid and discontinuous material property variations. One important example in mind is the simulation of blood flow in medium-to-large veins in humans. Finally, we also discuss some peculiarities of the model regarding the loss of strict hyperbolicity and uniqueness. In particular we show an example in which the solution of the Riemann problem is non unique.

On the Use of Rigid Body Modes in the Deflated Preconditioned Conjugate Gradient Method

Large discontinuities in material properties, such as those encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the preconditioned conjugate gradient method. This paper considers the deflated preconditioned conjugate gradient method for solving such systems. Our deflation technique uses as the deflation space the rigid body modes of sets of elements with homogeneous material properties. We show that in the deflated spectrum the small eigenvalues are mapped to zero and no longer negatively affect the convergence. We justify our approach through mathematical analysis and show with numerical experiments on both academic and realistic test problems that the convergence of our DPCG method is independent of discontinuities in the material properties.

XFEM modeling of ultrasonic wave propagation in polymer matrix particulate/fibrous composites

A method for representing discontinuous material properties in a heterogeneous domain by the extended finite element method (XFEM) has been applied to study ultrasonic wave propagation in polymer matrix particulate/fibrous composites. Representative volume elements of the composite material microstructure were generated by the random sequential adsorption (RSA) algorithm, where level set fields represent the matrix/inclusion interfaces within the domain. The equations of motion were integrated explicitly in time with mass lumping on the nodal and enriched degrees of freedom. This method shows improved agreement of the wave attenuation coefficients (WAC) from experimental measurements of ultrasonic, longitudinal waves in particulate composites compared with analytical methods, especially for the high volume fraction. The WAC were also computed for fiber composites, including random and aligned fiber orientations. Our results suggest that fibers aligned with the direction of wave propagation produce the greatest amount of attenuation, where scattering is the dominant energy attenuation mechanism when the wavelength approaches the characteristic length of the microstructure.

Theory and experimental method to determine bonding strength envelope of bi‐material interface

PurposeIt is well known that stress singularity may exist at the edges of a bonded bi‐material interface due to the discontinuity of material properties. This stress singularity causes difficulty in accurately determining the bi‐material interface bonding strength. This paper aims to present a new design of specimen geometry to eliminate the stress singularity and present an experimental procedure to more accurately determine the bonding strength of the bi‐material interface.Design/methodology/approachThe design is based on an asymptotic analysis of the stress field near the free edge of bi‐material interface. The critical bonding angle, which delineates the singular and non‐singular stress field near the free edge, is determined.FindingsWith the new designed specimen and a special iterative calculation algorithm, the interface bonding strength envelope of an epoxy‐aluminum interface was experimentally determined.Originality/valueThis new design of specimen, experimental procedure and iterative algorithm may be applied to obtain more reasonable and accurate bonding strength data for a wide range of bi‐material interfaces.

An analytical model to determine the stress singularity and critical bonding angle for an elastic–viscoelastic bonded joint

Measurements of interface bonding strengths are necessary for predicting the failure behavior of structures and materials with bi-material interfaces. However, it is well known that due to the discontinuity of material properties, stress singularity may exist at the edges of the interface. For accurate determination of the bonding strength of bi-material interface, the elimination of the stress singularity is necessary. This paper presents an analytical solution for the determination of the stress singularity and the critical bonding angle of a bonded joint between elastic and viscoelastic materials. This solution is based on the analytical solution available for an elastic–elastic bonded joint via the elastic–viscoelastic corresponding principle. For the viscoelastic material, both time-independent and time-dependent Poisson’s ratios are considered to find its effect on the stress singularity. As an example, the developed solution is applied to a simulated aluminum-epoxy bonded joint with a spherical interface. It is found that the critical bonding angle and the order of the stress singularity are different for assuming a time-independent or time-dependent Poisson’s ratio of the idealized viscoelastic epoxy. With the analytical solution developed, it is possible to design an optimal interface geometry that can eliminate the stress singularity from the interface corner.