Discontinuous Material Properties Research Articles

Acoustic cloaking via a hybrid passive–active technique

Both passive and active methods have been employed extensively in recent times to create the effect of acoustic cloaking, with obstacles being made near-invisible over specific frequency ranges. Passive methods typically rely on metamaterials with complex microstructures, whereas active methods require an external power supply. Furthermore, a large number of active sources and sensors are typically required for active control. In this work, a hybrid method that combines passive and active approaches is developed. The passive acoustic cloak is designed theoretically via layered shells with alternating, functionally graded properties. The layer properties are informed by coupling transformation acoustics with homogenization theory and require unphysical values on the inner cloak wall. As such, some scattering occurs from the obstacle to be cloaked as well as due to the discontinuity in material properties across the shell boundaries. Using distributed monopole control sources, active control is then employed to nullify the non-zero scattered field outside the passive cloak domain. Effective cloaking is achieved using the hybrid passive–active control system.

The cell-centered Finite-Volume self-consistent approach for heterostructures: 1D electron gas at the Si–SiO2 interface

Achieving self-consistent convergence with the conventional effective-mass approach at ultra-low temperatures (below 4.2 K) is a challenging task, which mostly lies in the discontinuities in material properties (e.g. effective-mass, electron affinity, dielectric constant). In this article, we develop a novel self-consistent approach based on cell-centered finite-volume discretization of the Sturm–Liouville form of the effective-mass Schrödinger equation and generalized Poisson’s equation (FV-SP). We apply this approach to simulate the one-dimensional electron gas formed at the Si–SiO2 interface via a top gate. We find excellent self-consistent convergence from high to extremely low (as low as 50 mK) temperatures. We further examine the solidity of FV-SP method by changing external variables such as the electrochemical potential and the accumulative top gate voltage. Our approach allows for counting electron–electron interactions. Our results demonstrate that FV-SP approach is a powerful tool to solve effective-mass Hamiltonians.

Detection of grinding-induced defects in titanium matrix composites by polarized laser scattering

In high-speed grinding of TiC-reinforced titanium matrix composites (TiC-TMCs), grinding-induced defects, e.g., chinks and cracks, respectively caused by titanium matrix smearing and TiC fracturing, can reduce the fatigue life of a ground part. Since TiC-TMCs have heterogeneous and discontinuous material properties, it is difficult to detect the defects of a ground part by the traditional non-destructive methods. In this study, the polarized laser scattering (PLS) detection method was applied for the first time to detecting the grinding-induced defects of the additively manufactured (AMed) TiC-TMCs. Different surface morphologies were obtained at various grinding speeds. A correlation between the polarized laser scattering signal and the percentage of defects was revealed. With an increase in grinding speed, the surface and subsurface defects were reduced, and the PLS signal intensity was decreased correspondingly. This study provides a new non-destructive defect detection method for metal matrix composite parts.

A novel homogenization method for periodic piezoelectric composites via diffused material interface

In this article, we propose a novel homogenization method for piezoelectric composites with periodic microstructures. In the proposed homogenization method, one can find the analytical expressions for the material properties of the composites in the representative volume element (RVE) via diffused material interface method. The availability of analytical expressions for the material properties in the entire domain of the RVE enables one to determine the effective homogenized material properties with the standard finite element method. As the proposed homogenization method regularizes the discontinuity in material properties across the interfaces, there is no need to explicitly track the material interfaces while implementing the finite element method for determining the effective material properties. Hence, one can implement the proposed homogenized method for any piezoelectric composites with complicated material interfaces and determine the effective material properties. In this study, we have considered the piezoelectric composites to be comprised of periodic microstructures where the RVE constitutes a matrix with an inclusion of a specific shape. We have carried out a study on the effect of the shape of inclusion viz. square-shaped, I-shaped, T-shaped, and plus-shaped, and the size of inclusion on the effective piezoelectric material properties. We have also studied the influence of shape and size of inclusion on the electro-mechanical response of a homogenized piezoelectric continuum. To implement the proposed homogenization method, we have used Gridap, an open-source finite element toolbox in Julia that provides very compact codes freely available to all the researchers and makes a third-party verification of the proposed method straightforward.

Modeling diffusion and types I-V sorption of water vapor in heterogeneous systems

A sorption and diffusion model is developed using a mobile-immobile decomposition of the vapor concentration. Henry’s mode represents the mobile species which diffuses through the solid, while Langmuir and pooling modes represent possible immobilization processes that cause nonlinearities in sorption capacity. The model can simulate the five classical sorption types and can incorporate additional dynamics compared to models based on thermodynamic equilibrium. Additionally, the framework allows for discontinuities in material properties encountered in multi-material systems. The resulting non-linear coupled equations are solved by employing a finite element method to discretize in space, and a backward Euler method to discretize in time. The discretized system of equations is solved via a Newton-type iteration scheme at each time-step. Four different materials are parameterized for the model, which are then used in examples to demonstrate the model’s ability to capture complex sorption processes in multi-material systems.

Proper orthogonal decomposition based seismic source wavefield reconstruction for finite element reverse time migration

The large storage requirement is a critical issue in cross-correlation imaging-condition based reverse time migration (RTM), because it requires the operation of the source and receiver wavefields at the same time. The boundary value method (BVM), based on the finite difference method (FDM), can be used to reconstruct the source wavefield in the reverse time propagation in the same way as the receiver wavefield, which can reduce the storage burden of the RTM data. Considering that the FDM cannot well handle models with discontinuous material properties and rough interfaces, we develop a source wavefield reconstruction strategy based on the finite element method (FEM), using proper orthogonal decomposition (POD) to enhance computational efficiency. In this method, we divide the whole time period into several segments, and construct the POD basis functions to get a reduced order model (ROM) for the source wavefield reconstruction in each segment. We show the corresponding quantitative analysis of the storage requirement of the POD-FEM. Numerical tests on the homogeneous model show the effectiveness of the proposed method, while the layered model and part of the Marmousi model tests indicate that the POD-FEM can keep an excellent balance between computational efficiency and memory usage compared with the full-stored method (FSM) and the BVM, and can be effectively applied in imaging.

A hybrid pressure approximation in the control volume finite element method for multiphase flow and transport in heterogeneous porous media

We present a new hybrid pressure formulation applied to the control volume finite element (CVFE) method to model multiphase flow and transport in highly heterogeneous porous media. The formulation effectively captures sharp saturation changes in the presence of discontinuous material properties by employing a discontinuous pressure approximation at material interfaces. The heterogeneous porous medium is divided into sub-domains within which material properties are uniform or smoothly varying. By construction, the resultant control volume dual mesh is restricted within a sub-domain. The artificial mass leakage across material property boundaries observed in classical CVFE methods is therefore circumvented. The approach applies the robust continuous pressure approximation in the rest of the computational domain; the discontinuous approximation is applied only at the sub-domain boundaries. The discontinuous parameters necessary to achieve mass conservative solutions, locally and globally, are described. We demonstrate the accuracy and efficiency of the new approach by comparison with the classical continuous CVFE method on various examples of heterogeneous domains as well as establishing the convergence of the numerical method. The proposed hybrid formulation significantly outperforms the accuracy and efficiency of classical CVFE methods that use the same order of approximation for modeling multiphase flow in heterogeneous porous media.

Non-iterative domain decomposition for the Helmholtz equation with strong material discontinuities

Many wave propagation problems involve discontinuous material properties. We propose to solve such problems by non-overlapping domain decomposition combined with the method of difference potentials (MDP). The MDP reduces the Helmholtz equation on each subdomain to a Calderon's boundary equation with projection on the boundary. The unknowns for the Calderon's equation are the Dirichlet and Neumann data. Coupling between neighboring subdomains is rendered by applying their respective Calderon's equations to the same data at the common interface. Solutions on individual subdomains are computed concurrently using a direct solver. Our method proves to be insensitive to large jumps in the wavenumber for transmission problems, as well as interior cross-points and mixed boundary conditions, which may be a challenge to many other domain decomposition methods.

Interaction integral method for crack-tip intensity factor evaluations of magneto-electro-elastic materials with residual strain

Magneto-electro-elastic (MEE) materials generally consist of piezoelectric and piezomagnetic constituent phases so that the residual strain often occurs during the manufacturing process. The distributed residual strain and complex interfaces between constituents cause a great challenge to the fracture analysis of MEE materials. Considering the effect of the residual strain, this paper develops an interaction integral (I-integral) method for the extraction of the fracture parameters of an impermeable crack in nonhomogeneous MEE materials. The I-integral method has several merits: 1) it can decouple stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF); 2) the I-integral is independent of the size of the integration domain, even when the integration domain contains nonhomogeneous and discontinuous material properties. Combined with the extended finite element method (XFEM), the I-integral is adopted to investigate several crack problems subjected to homogeneous, nonhomogeneous and discontinuous residual strains. The I-integral values calculated by different integration domains agree well with each other (relative deviation < 1%) for nonhomogeneous and discontinuous MEE properties. Then, single-crack and multi-crack problems are studied to show the influences of residual strain distributions on the intensity factors. Homogeneous residual strain affects the SIFs dramatically but has a slight influence on the EDIF and MIIF. By comparison, nonhomogeneous residual strains affect the SIFs, EDIF and MIIF significantly. In addition, the residual strain gradients affect the variations of intensity factors with crack location extremely.

Zigzag Theory for Piezoelectric-Layer-Integrated Functionally Graded Material Plates

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

Impact of Numerical Methods in Thermal Modeling of Li‐Ion Batteries on Temperature Distribution and Computation Time

Thermal battery modeling is important for further battery development and optimization. The temperature strongly influences the performance and aging behavior. In the cell stack, electrochemical processes take place resulting in a large amount of heat release, which, in turn, affects the temperature distribution. Therefore, the main focus is on the cell stack, the most complex structure inside the cell. In particular, the discontinuous and anisotropic material properties represent a major challenge for simulations due to the layering. This work proposes self‐developed methods, based on the Finite Volume Method and the Finite Element Method, taking on these challenges. First, for both methods the functionality is verified and numerical convergence is validated. These, and also classical methods, are compared based on test problems with a known analytical solution in view of numerical errors as well as computing time. It if found that their accuracy and efficiency depends strongly on the specific problem, which makes their numerical investigation necessary and inevitable. Second, the methods are evaluated on a specific battery problem. Their results are plausible and correspond to the physical phenomena.

On the effective behavior of viscoelastic composites in three dimensions

We address the calculation of the effective properties of non-aging linear viscoelastic composite materials. This is done by solving the microscale periodic local problems obtained via the Asymptotic Homogenization Method (AHM) by means of finite element three-dimensional simulations. The work comprises the investigation of the effective creep and relaxation behavior for a variety of fiber and inclusion reinforced structures (e.g. polymeric matrix composites). As starting point, we consider the elastic-viscoelastic correspondence principle and the Laplace-Carson transform. Then, a classical asymptotic homogenization approach for composites with discontinuous material properties and perfect contact at the interface between the constituents is performed. In particular, we reach to the stress jump conditions from local problems and obtain the corresponding interface loads. Furthermore, we solve numerically the local problems in the Laplace-Carson domain, and compute the effective coefficients. Moreover, the inversion to the original temporal space is also carried out. Finally, we compare our results with those obtained from different homogenization approaches, such as the Finite-Volume Direct Averaging Micromechanics (FVDAM) and the Locally Exact Homogenization Theory (LEHT).

Systematic development of upwind numerical fluxes for the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media with physical interfaces

This research work presents, within a unified and wave oriented variational framework, systematic development of upwind numerical fluxes for the space discontinuous Galerkin methods to model elastic wave propagation in multidimensional anisotropic media with discontinuous material properties. Both first-order velocity–stress and velocity–strain wave formulations are considered. The proposed approach allows the derivation of upwind numerical fluxes in a well structured and hierarchical way according to the degree of inhomogeneity across a physical interface. The numerical fluxes that are exact solutions of a relevant Riemann problem defined at a physical interface are obtained. The developed explicit and intrinsic tensorial expressions of upwind numerical fluxes in multidimensional case allow a better understanding and analysis of the physical meaning of involved terms. As numerical applications, an example with a physical interface separating two materials, one anisotropic and the other isotropic, and an example of polycrystalline material that presents a particular case with a larger number of physical interfaces, are considered. The proposed numerical fluxes are numerically investigated and validated.

A new domain-independent interaction integral for three-dimensional non-homogeneous thermoelastic materials

This work establishes a new domain-independent interaction integral for the evaluation of mixed-mode stress intensity factors (SIFs) in three-dimensional non-homogeneous materials with complex thermo-mechanical interfaces subjected to thermo-mechanical loadings. A new auxiliary field called Zero-Mean stress field is developed to deal with the mismatch of thermoelastic properties across the material interfaces. An area integral along with the material interfaces, which is indispensable to keep domain-independence of the conventional interaction integral using Williams’ auxiliary fields, is eliminated by using this new auxiliary. A crack-face integral is added to maintain the domain-independence of this new method. The expression of this new domain-independent interaction integral contains no thermal property parameters. Because of this feature, the new interaction integral holds domain-independent for both discontinuous mechanical properties and discontinuous thermal properties. Since tracking crack-planes is much easier than tracking a material interface in numerical implementations, it brings convenience to the practical applications to analyze three-dimensional non-homogeneous materials with discontinuous thermoelastic material properties. The new interaction integral, combined with the finite element method, is applied to investigate several representative examples, which demonstrates that the new interaction integral has good accuracy and domain-independence for thermo-mechanical interfaces.

Direct Method Solution of 3-D Magnetotelluric Modeling Using Vector Finite Element Method

As exploration is forced into more difficult areas with complex three-dimensional (3-D) structural environments, the importance of 3-D magnetotelluric (MT) modeling is essential for the correct interpretation of MT data. To reduce the complexity of the calculations introduced by 3-D models, iterative forward calculation of MT response functions is used as basis for inversion of 3-D MT data. This paper proposes an alternative procedure for making reliable 3-D MT modeling codes for forward calculation that is not only effective but also accurate. This is accomplished by using a direct method to solve the linear systems arising from the discretization process in the vector finite element approach. The vector finite element method is known for its capability of overcoming difficulties in modeling caused by possible jumps of normal components across discontinuities of material properties. Meanwhile, by using a direct method rather than an iterative method, the process of solving the linear equations does not suffer from slow convergence. Here, we present a comparison between our modeling codes and codes based on a different approach. In the resulting 3-D MT responses it was found that the proposed method has high accuracy.

Some comparisons and analyses of time or space discontinuous Galerkin methods applied to elastic wave propagation in anisotropic and heterogeneous media

This research work presents some comparisons and analyses of the time discontinuous space–time Galerkin method and the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media. Mechanism of both methods to ensure their stability using time or space discontinuities of unknown fields is analyzed and compared. The most general case of anisotropic and heterogeneous media with physical interfaces of discontinuous material properties is considered, especially for the space discontinuous Galerkin method. A new stability result is proved. Numerical applications to different elastic media, more particularly polycrystalline materials containing a large number of physical interfaces, are also presented to confirm theoretical analyses.

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

Fatigue damage analysis of offshore wind turbine monopile weldments

Offshore wind turbines (OWT) are subjected to harsh environmental conditions in addition to the variable service loads. The present study is aimed at performing a realistic fatigue life estimation of the monopile structure using operational service loads recorded by online monitoring systems. Fatigue damage analysis has been conducted at the circumferential weld joints using finite element (FE) method by considering geometrical and material property discontinuities. Global-local modelling of the OWT was performed in as-welded condition to capture the local stress range at the weld toe, which acts as the critical site where cracks are most likely to initiate and propagate. The S-N fatigue design approach and maximum stress range at the weld toe have been used to determine the fatigue crack initiation life in monopiles. The results from the proposed approach show that a realistic life assessment can be made on monopile structures by accounting for the geometrical effects at the circumferential welds.

A high-order accurate five-equations compressible multiphase approach for viscoelastic fluids and solids with relaxation and elasticity

A novel Eulerian approach is proposed for numerical simulations of wave propagation in viscoelastic media, for application to shocks interacting with interfaces between fluids and solids. We extend the five-equations multiphase interface-capturing model, based on the idea that all the materials (gases, liquids, solids) obey the same equation of state with spatially varying properties, to incorporate the desired constitutive relation; in this context, interfaces are represented by discontinuities in material properties. We consider problems in which the deformations are small, such that the substances can be described by linear constitutive relations, specifically, Maxwell, Kelvin–Voigt or generalized Zener models. The main challenge lies in representing the combination of viscoelastic, multiphase and compressible flow. One particular difficulty is the calculation of strains in an Eulerian framework, which we address by using a conventional hypoelastic model in which an objective time derivative (Lie derivative) of the constitutive relation is taken to evolve strain rates instead. The resulting eigensystem is analyzed to identify wave speeds and characteristic variables. The spatial scheme is based on a solution-adaptive formulation, in which a discontinuity sensor discriminates between smooth and discontinuous regions. To compute the convective fluxes, explicit high-order central differences are applied in smooth regions, while a high-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme is used at discontinuities (shocks, material interfaces and contacts). The numerical method is verified in a comprehensive fashion using a series of smooth and discontinuous (shocks and interfaces), one- and two-dimensional test problems.

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.