Finite Volume Element Research Articles

Hemocompatibile Thin Films Assessed under Blood Flow Shear Forces.

The aim of this study was to minimize the risk of life-threatening thromboembolism in the ventricle through the use of a new biomimetic heart valve based on metal–polymer composites. Finite volume element simulations of blood adhesion to the material were carried out, encompassing radial flow and the cone and plane test together with determination of the effect of boundary conditions. Both tilt-disc and bicuspid valves do not have optimized blood flow due to their design based on rigid valve materials (leaflet made of pyrolytic carbon). The main objective was the development of materials with specific properties dedicated to contact with blood. Materials were evaluated by dynamic tests using blood, concentrates, and whole human blood. Hemostability tests under hydrodynamic conditions were related to the mechanical properties of thin-film materials obtained from tribological tests. The quality of the coatings was high enough to avoid damage to the coating even as they were exposed up to maximum loading. Analysis towards blood concentrates of the hydrogenated carbon sample and the nitrogen-doped hydrogenated carbon sample revealed that the interaction of the coating with erythrocytes was the strongest. Hemocompatibility evaluation under hydrodynamic conditions confirmed very good properties of the developed coatings.

Lattice Boltzmann simulation of the ERCOFTAC pump impeller

The analysis of unsteady flows is becoming more and more important in the field of hydraulic turbomachinery applications, both for stable and especially for unstable operating conditions and other transient phenomena. Unsteady Reynolds-averaged Navier-Stokes (RANS) methods for incompressible flows, implemented via finite volume or finite element approaches, reach their limit in both accuracy and acceptable computational time. The accuracy mainly suffers due to the strong dependence of the solution on turbulence model variants. Furthermore, incompressibility of the flow and the linked implicit pressure equation makes large eddy simulations (LES) of Navier-Stokes-based methods for such flows prohibitively expensive. An alternative to Navier-Stokes fluid solvers are methods based on the solution of the discrete Boltzmann equation, so-called Lattice Boltzmann Methods (LBM). The purpose of this work is to present first results of an ongoing project involving the simulation of unsteady hydraulic turbomachines at high Reynolds numbers. For this purpose, the ERCOFTAC pump impeller was selected as a test case. Results obtained with the current implementation of the in-house code suggest, that the present methodology could provide a viable alternative for the analysis of unsteady flow phenomena in hydromachines.

A new high order finite volume element solution on arbitrary triangular and quadrilateral meshes

In this work, for the two dimensional anisotropic diffusion problem, the classical kth order finite element solution is postprocessed to obtain a new finite volume element solution, such that the new solution satisfies the local conservation property on a certain dual mesh, and converges to the analytic solution with optimal rates. The postprocessing algorithm has a local nature and can be conducted element by element. The novelty of this paper is the introduction of a new bubble function, which enables us to prove the existence and uniqueness of the postprocessed solution on arbitrary triangular or convex quadrilateral meshes with full anisotropic diffusion tensor. The theoretical findings are also verified by some numerical results.

Existence of Dissipative Solutions to the Compressible Navier-Stokes System with Potential Temperature Transport

We introduce dissipative solutions to the compressible Navier-Stokes system with potential temperature transport motivated by the concept of Young measures. We prove their global-in-time existence by means of convergence analysis of a mixed finite element-finite volume method. If a strong solution to the compressible Navier-Stokes system with potential temperature transport exists, we prove the strong convergence of numerical solutions. Our results hold for the full range of adiabatic indices including the physically relevant cases in which the existence of global-in-time weak solutions is open.

Stress average rule derived through the principle of virtual power

AbstractStress and strain average rules are the key conceptual pillars of the wide field of continuum micromechanics of materials. The aforementioned rules express that the spatial average of (micro‐)stress and (micro‐)strain fields throughout a microscopically finite representative volume element (RVE) are equal to the (macro‐)stress and (macro‐)strain values associated with the corresponding macroscopically infinitesimal volume element (macroscopic material point). According to the famous contribution of Hashin, stress and strain average rules are derived from equilibrium and compatibility conditions, together with (micro‐)displacement and (micro‐)traction boundary conditions associated with homogeneous (macro‐)strains and (macro‐)stresses, respectively. However, as, strictly speaking, only displacements or tractions can be described at the boundary, the remaining average rule turns out as a mere definition. We here suggest a way to do without such a definition, by resorting to the principle of virtual power (PVP) as a means to guarantee mechanical equilibrium: at the boundary of the RVE, we prescribe virtual (micro‐)velocities, which are linked to arbitrary, but homogeneous virtual (macro‐)velocities and (macro‐)strain rates, while the latter are also linked, in a multilinear fashion, with the microscopic virtual strain rate fields inside the RVE. Considering, under these conditions, equivalence of the macroscopic and the microscopic expressions for the virtual power densities of the internal and the external forces yields the well‐known stress average rule and, in case of microscopically uniform force fields, a volume force average rule. The same strategy applied to an RVE hosting single forces between atomistic mass points, readily yields the macroscopic “internal virial stress tensor.”

Analysis of a finite volume element scheme for solving the model two-phase nonequilibrium flow problem

The paper proposes a hybrid numerical method for solving a model problem of two-phase nonequilibrium flow of an incompressible fluid in a porous medium. This problem is relevant in the modern theory of the motion of multiphase fluids in porous media and has many applications. The studied model is based on the assumption that the relative phase permeabilities and capillary pressure depend not only on saturation, but also on its time derivative. The saturation equation in this problem refers to the type of convection-diffusion with a predominance of convection, which also includes a third-order term to account for the nonequilibrium effects. Due to the hyperbolic nature of the equation, its solution is accompanied by a number of difficulties that lead to the need for an appropriate choice of the solution method. In contrast to previous works, this paper uses a finite volume element method for solving the problem, the construction of which is based on integral balance equations, and an approximate solution is chosen from the finite element space. To discretize the problem, two different dual grids are used based on the main triangulation. In this paper, a number of a priori estimates are obtained which yields the unconditional stability of the scheme as well as its convergence with the second order. The advantages of the approach used include the local conservatism of the scheme, as well as the comparative simplicity of the software implementation of the method. These results are confirmed by a numerical test carried out on the example of a model problem.

A novel monotone finite volume element scheme for diffusion equations

We present a novel monotone finite volume element scheme for the diffusion problem on triangular grids. Firstly, one fictitious triangular element is introduced for each edge of any element, and another vertex of this fictitious element is on the outward extended straight line passing through two points: The vertex opposite to this edge and the barycenter of the element. The diffusive tensor and the solution in this element are continuously extended to its corresponding fictitious element. The fictitious element and the continuous extension not only avoid the phenomenon that the control volume segment passes through other elements, but also are helpful to the construction and theoretical analysis for our monotone scheme. Secondly, one new nonlinear two-point flux formulation is obtained for some part control volume edge whose another endpoint is the barycenter of the fictitious element, and this straight segment crosses through the common edge of the triangular element and its fictitious one. The approximation of κ∇u on this straight control volume edge is constructed by some weighted average with the gradients of two finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we proved that the new scheme is monotone under some conditions: the diffusion tensor and exact solution functions are properly smooth, and the partition step size h is properly small, and the solutions on three nodes of any element are not nonnegative, and furthermore not simultaneously zero, and the nodes of the fictitious element are properly chosen. We also proved that the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and confirm its accuracy and stability.

Biomechanics of human trabecular meshwork in healthy and glaucoma eyes via dynamic Schlemm's canal pressurization

Background and objectiveThe trabecular meshwork (TM) consists of extracellular matrix (ECM) with embedded collagen and elastin fibers providing its mechanical support. TM stiffness is considerably higher in glaucoma eyes. Emerging data indicates that the TM moves dynamically with transient intraocular pressure (IOP) fluctuations, implying the viscoelastic mechanical behavior of the TM. However, little is known about TM viscoelastic behavior. We calculated the viscoelastic mechanical properties of the TM in n = 2 healthy and n = 2 glaucoma eyes. MethodsA quadrant of the anterior segment was submerged in a saline bath, and a cannula connected to an adjustable saline reservoir was inserted into Schlemm's canal (SC). A spectral domain-OCT (SD-OCT) provided continuous cross-sectional B-scans of the TM/JCT/SC complex during pressure oscillation from 0 to 30 mmHg at two locations. The TM/JCT/SC complex boundaries were delineated to construct a 20-µm-thick volume finite element (FE) mesh. Pre-tensioned collagen and elastin fibrils were embedded in the model using a mesh-free penalty-based cable-in-solid algorithm. SC pressure was represented by a position- and time-dependent pressure boundary; floating boundary conditions were applied to the other cut edges of the model. An FE-optimization algorithm was used to adjust the ECM/fiber mechanical properties such that the TM/JCT/SC model and SD-OCT imaging data best matched over time. ResultsSignificantly larger short- and long-time ECM shear moduli (p = 0.0032), and collagen (1.82x) and elastin (2.72x) fibril elastic moduli (p = 0.0001), were found in the TM of glaucoma eyes compared to healthy controls. ConclusionsThese findings provide additional clarity on the mechanical property differences in healthy and glaucomatous outflow pathway under dynamic loading. Understanding the viscoelastic properties of the TM may serve as a new biomarker in early diagnosis of glaucoma.

Integration of Electromagnetic Geophysics Forward Simulation in Coupled Flow and Geomechanics for Monitoring a Gas Hydrate Deposit Located in the Ulleung Basin, East Sea, Korea

We investigate the feasibility of electromagnetic (EM) geophysics methods to detect the dissociation of gas hydrate specifically from a gas hydrate deposit located in the Ulleung Basin, East Sea, Korea via an integrated flow-geomechanics-EM geophysics simulation. To this end, coupled flow and geomechanics simulation is first performed with the multiple porosity model employed, where a mixed formulation with the finite volume (FV) and finite element (FE) methods are taken for the flow and geomechanics, respectively. From the saturation and porosity fields obtained from the coupled flow and geomechanics, the electrical conductivity model is established for the EM simulation. Solving the partial differential equation of electrical diffusion which is linearized using the 3D finite element method (FEM), the EM fields are then computed. For numerical experiments, particularly two approaches in the configuration for the EM methods are compared in this contribution: the surface-to-surface and the surface-to-borehole methods. When the surface-to-surface EM method is employed, the EM is found to be less sensitive, implying low detectability. Especially for the short term of production, the low detectability is attributed to the similarity of electrical resistivity between the dissociated gas (CH4) and hydrate as well as the specific dissociation pattern within the intercalated composites of the field. On the other hand, when the surface-to-borehole EM method is employed, its sensitivity to capture the produced gas flow is improved, confirming its detectability in monitoring gas flow. Hence, the EM geophysics simulation integrated with coupled flow and geomechanics can be a potential tool for monitoring gas hydrate deposits.

Numerical Modeling of Flexible Net Panels under Steady Flow Using a Coupled Fluid–Structure Partitioned Scheme

Fluid–structure interactions of flexible net panels are complex and lack sufficient exploration. To examine the flow characteristics of flexible net panels with large deformation, we propose a partitioned coupling scheme in this paper. The coupled fluid–structure equations are solved separately under finite volume and finite element frameworks. The interface traction from the fluid solver is considered as a Neumann boundary condition for the solid domain, and the interface velocity is applied as a Dirichlet boundary condition for the fluid problem. Then, the forces can be transferred along the interface via Dirichlet-to-Neumann mapping. The results show that both the drag coefficient and the velocity reduction increase alongside the net solidity ratio (Sn), but they decrease as the Reynolds number/attack angle increases. A comparative study of drag coefficients is made between the present numerical simulations and the analytical predictions. This paper also examines the velocity distribution and vortex formation of flexible net panels. A single vortex forms in the shear layers and the wake when Sn = 0.16, and a pair of vortices mostly forms in the wake when Sn = 0.33. The vertical net twines predominantly affect the formation of the vortex behind the net, leading to delayed vortex shedding. The flow exhibits wake interactions due to the interference between the net twines in the high-solidity net panel. No such interference occurs in the low-solidity net panel, but the altered shear layers could cause severe velocity fluctuations in the near field.

Parallel domain decomposition schemes based on finite volume element discretization for nonsteady-state diffusion equations on distorted meshes

In this paper, several domain decomposition algorithms are considered to solve parallelly non-steady diffusion equations. The contributions of this work are threefold. The first one is that we present effective parallel domain decomposition schemes underlying the classical and monotone finite volume element discretization. These schemes adopt a three-stage technique, which is also used in the difference method and the finite volume method. Positivity of the parallel scheme based on monotone finite volume element method is proved. The second one is that we propose a new domain decomposition computational strategy to improve the correction step in prediction-correction frame such that the whole parallel process is simplified. The third one is that we consider a parallel algorithm build on quadratic finite volume element method, which preserves local conservation and has higher convergence rates. These parallel schemes are of intrinsic parallelism since we apply the prediction-correction technique to the interface of subdomains coming from an artificial division of computational domain or multi-medium physical domain. The proposed parallel schemes need only local communication among neighboring processors. Numerical results are presented to illuminate the accuracy, stability and parallelism of the parallel schemes.

Matrices associated to two conservative discretizations of Riesz fractional operators and related multigrid solvers

AbstractIn this article, we focus on a two‐dimensional conservative steady‐state Riesz fractional diffusion problem. As is typical for problems in conservative form, we adopt a finite volume (FV)‐based discretization approach. Precisely, we use both classical FVs and the so‐called finite volume elements (FVEs). While FVEs have already been applied in the context of fractional diffusion equations, classical FVs have only been applied in first‐order discretizations. By exploiting the Toeplitz‐like structure of the resulting coefficient matrices, we perform a qualitative study of their spectrum and conditioning through their symbol, leading to the design of a second‐order FV discretization. This same information is leveraged to discuss parameter‐free symbol‐based multigrid methods for both discretizations. Tests on the approximation error and the performances of the considered solvers are given as well.

Coupled thermal-hydrodynamic-mechanical–chemical numerical simulation for gas production from hydrate-bearing sediments based on hybrid finite volume and finite element method

Gas production from hydrates induced by depressurization is a complex thermal-hydrodynamic-mechanical–chemical (THMC) coupled process. In this paper, we present a THMC coupled model to simulate the fluid flow in hydrate-bearing sediments (HBS) and the geomechanical behavior of HBS. The model is made of two subsystems, which are the fluid part of non-isothermal multi-phase flow with hydrate kinetic and solid part of geomechanical deformation. It accounts for two-way coupling effects between these two subsystems, i.e. the effect of pore pressure and hydrate dissociation on the solid mechanical behavior and the effect of stress on the hydraulic behavior. A new numerical method based on the hybrid control volume finite element method (CVFEM)-finite element method (FEM) is developed to solve the mathematical models. The local conservative CVFEM is used for the fluid part, and the standard FEM for the solid part. In the framework of hybrid CVFEM-FEM, the local conservation is reserved and the primary variables for the two subsystem are co-located. A multi-point flux approximation (MPFA) is adopted without orthogonal meshes so that it is very flexible to build complex geometrical models. The accuracy and reliability of the newly developed simulator QIMGHyd-THMC are tested by comparing with two experimental examples and a large-scale benchmark problem of other popular simulators.

Multiscale modelling framework for elasticity of ultra high strength concrete using nano/microscale characterization and finite element representative volume element analysis

Ultra-High Strength Concrete (UHSC) (greater than 100 MPa) is a mechanically superior material compared with the Normal Strength Concrete (NSC) due to its inherent performance characteristics. Improved modulus of elasticity is one of the key target characteristics in the development of UHSC. Macroscopic response of UHSC is a result of a multitude of phases in different spatial length scales such as mesoscale, microscale, nanoscale etc. and investigating these spatial scales can yield a better understanding about contribution of each heterogenous phases to the macroscopic behaviour of UHSC. In this paper, a new multiscale modelling method including procedures to obtain micro/nano scale properties is proposed to predict the macroscopic elastic modulus using nanoindentation experiments, hydration simulations, Scanning Electron Microscopy (SEM) and Finite Element Representative Volume Element (FE-RVE) modelling.Characterization of nanomechanical properties of the cementitious composite was carried out using nanoindentation, microstructural characterization was performed using scanning electron microscopy, and hydration simulation of the cementitious paste was carried out using Virtual Cement and Concrete Testing Laboratory (VCCTL) software. A five-level multiscale framework is proposed for UHSC and results from these experimental testing and simulations were used as inputs in the proposed framework.A novel algorithm which can model any volume fraction of different phases was developed to generate geometries for RVEs to be used in FE-RVE simulations. Upscaling of elastic modulus using FE-RVE was found to be very accurate, and this method can generate detailed variation of microfields inside the RVE. A parametric study was carried out on how varying inhomogeneities in the RVE, boundary conditions, and the shape of the inhomogeneities would affect the homogenized elastic modulus.Continuum micromechanics models such as Mori-Tanaka method and Self Consistent Scheme were used for the analytical homogenization at each scale for comparison with FE-RVE method. The results of the proposed FE-RVE analysis, the Mean Field Homogenization (MFH) method, and experiment were compared and found to be a very good fit.

Thermomechanical simulations of the transient coupled effect of thermal cycling and oxidation on residual stresses in thermal barrier coatings

Failures in thermal barrier coatings (TBCs) are associated with the build-up of residual stresses that result from thermal cycling, growth strain, and stress relaxation associated with high temperatures. To address these highly coupled processes, three aspects were examined. The first was concerned with the effect of thermal cycling and thermal gradients on the resulting residual stress fields. The second with the dynamic growth of thermally grown oxide (TGO) layer using novel finite volume-finite element algorithms. In the third, we examined the effect of stress relaxation on the (TC/TGO) interface. We modelled these highly coupled processes using transient thermomechanical finite element simulations. The temperature profile and state of oxidation variation with time were imported as a predefined field and solved in ANSYS nonlinear platform. Our results revealed that stress relaxation of the TGO stresses at high temperatures leads to a reduction in the TC/TGO interfacial stresses. They also revealed that the use of the isotropic hardening rule limits the increase in plastic deformation of the bond coat (BC), while the use of kinematic hardening rule leads to ratcheting. Furthermore, we highlighted the importance of considering uneven growth of TGO on the resulting stress field.

Two-grid finite volume element method for the time-dependent Schrödinger equation

In this paper, we construct a backward Euler full-discrete two-grid finite volume element scheme for solving time-dependent Schrödinger equation. Combining the idea of the two-grid discretization, the original coupling system is solved in the coarse grid space, and the decoupling system with two independent Poisson problems is solved in the fine grid space, which ensures the accuracy and improves the computational efficiency. We further prove the optimal error estimate of the scheme rigorously. Finally, the numerical simulation results show that the two-grid method is more effective than the standard finite volume element method in solving coupled partial differential problems.

Finite volume element methods for two-dimensional time fractional reaction–diffusion equations on triangular grids

In this paper, time fractional reaction–diffusion equations with the Caputo fractional derivative are solved by using the classical -formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability results and optimal a priori error estimate in -norm are derived, but it is difficult to obtain the corresponding results in -norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness.

Vari-stiffness characteristics of a 3D SMA hybrid basalt woven composite

Shape Memory Alloy Hybrid Composite (SMAHC) obtained by weaving shape memory alloy into composite materials is an intelligent composite material with broad application prospects. 3D woven composites can provide stronger delamination resistance, and taking SMA as warp to manufacture SMAHC is conducive to reducing bending and giving full play to the performance of SMA. In this study, SMA was woven into 3D woven as stuffer warp, and orthogonal experimental design was carried out using amount of SMA, pre-strain of the SMA and the fabric structure as three independent variables to investigate how to maximize the stiffness control effect of the material. A finite element Representative volume element (RVE) model was established to predict local responses. Water transfer print (WTP) and digital imagination correlation (DIC) technique were used to exclude temperature disturbances and monitor the strain field changes during the tensioning process. The results show that SMA has a greater effect on the surrounding yarn than expected. The influence of three factors on the composite structure is also discussed in detail and weaving suggestions for 3D SMAHC are put forward.

A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

Two-Grid Immersed Finite Volume Element Methods for Semi-Linear Elliptic Interface Problems with Non-Homogeneous Jump Conditions

Two-Grid Immersed Finite Volume Element Methods for Semi-Linear Elliptic Interface Problems with Non-Homogeneous Jump Conditions