Anisotropic Mesh Research Articles

Impact of Anisotropic Mesh Adaptation on the Aerothermodynamics of Atmospheric Reentry

The presence of complex geometries and/or multiple bodies during atmospheric reentry may lead to complex and directional flow features, such as shock waves and shear layers, that need to be correctly predicted to ensure accurate calculation of the aerothermal loads during reentry. Central in ensuring reliable numerical prediction of loads is the adoption of meshes that ensure grid independence and minimize the misalignment between the directional flow features and the grid cell interfaces, a situation that is known to give rise to nonphysical behaviors and spurious oscillations. The use of anisotropic unstructured grid adaptation is here presented as a means to ensure appropriate grid resolution and alignment with directional flow features for cases where the use of structured grids is not always possible or practical. Results highlight the effectiveness and reliability of anisotropic mesh adaptation in successfully predicting the location of shock discontinuities as well as surface aerothermodynamic quantities, while providing results comparable with established approaches relying on structured meshes. Results are presented for single- and multiple-body cases through comparison with experimental data and reference numerical solutions.

On mesh refinement procedures for the virtual element method for two-dimensional elastic problems

The virtual element method is an extension of the finite element method permitting arbitrary polygonal/polyhedral element geometry. The method’s mesh flexibility is well-known and is beginning to be exploited in the context of mesh adaptivity. In this work a variety of novel approaches to the computation of isotropic and anisotropic mesh refinement indicators suited to the VEM are presented and comparatively assessed for the case of two-dimensional linear elasticity on structured meshes. Additionally, a novel investigation of the contributions of the two distinctive terms of the VE matrix, the consistency and stabilization terms, in the context of mesh adaptivity is performed. The results demonstrate that the best performance is achieved using combination of refinement procedures based on the displacement and strain fields. Furthermore, the results indicate that the stabilization term can be exploited to enhance adaptive refinement procedures.

Implicitly Extrapolated Geometric Multigrid on Disk-Like Domains for the Gyrokinetic Poisson Equation from Fusion Plasma Applications

The gyrokinetic Poisson equation arises as a subproblem of Tokamak fusion reactor simulations. It is often posed on disk-like cross sections of the Tokamak that are represented in generalized polar coordinates. On the resulting curvilinear anisotropic meshes, we discretize the differential equation by finite differences or low order finite elements. Using an implicit extrapolation technique similar to multigrid tau -extrapolation, the approximation order can be increased. This technique can be naturally integrated in a matrix-free geometric multigrid algorithm. Special smoothers are developed to deal with the mesh anisotropy arising from the curvilinear coordinate system and mesh grading.

An efficient high-order numerical solver for diffusion equations with strong anisotropy

In this paper, we present an interior penalty discontinuous Galerkin finite element scheme for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. We demonstrate the accuracy produced by the high-order scheme and develop an efficient preconditioning technique to solve the corresponding linear system, which is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the accuracy and efficiency of the proposed algorithm.

Enhanced anisotropic block-based adaptive mesh refinement for three-dimensional inviscid and viscous compressible flows

A parallel anisotropic block-based adaptive mesh refinement (AMR) algorithm is proposed for the prediction of physically complex flow problems having disparate spatial and temporal scales and exhibiting highly anisotropic features on three-dimensional multi-block body-fitted hexahedral meshes with non-uniform grid blocks. The proposed AMR scheme makes use of a binary tree hierarchical data structure to permit anisotropic refinement of grid blocks in a preferred coordinate direction as dictated by appropriately selected physics-based refinement criteria. The anisotropic coarsening of the grid blocks in a manner that is independent of the refinement history allows the mesh to rapidly re-adapt for evolving unsteady flow applications. Moreover, the proposed anisotropic AMR scheme adopts a non-uniform representation of the cells within each block by directly using the neighboring cells as the ghost cells, even at a grid resolution change. This affords a number of computational advantages especially related to evaluating the solution in the ghost cells as well as to ensuring flux conservation at block interfaces. A modified upwind-based finite-volume spatial discretization scheme is applied in conjunction with the AMR scheme to the solution of Euler and Navier-Stokes equations for inviscid and viscous compressible gaseous flows. Steady-state and time-varying flow problems are considered on anisotropic adapted meshes. The potential flexibility and efficiency of the proposed enhanced anisotropic AMR scheme are demonstrated for the simulation of a number of representative flows of varying complexity.

Design of cellular materials for multiscale topology optimization: application to patient-specific orthopedic devices

A flexible problem-specific multiscale topology optimization is introduced to associate different areas of the design domain with diverse microstructures extracted from a dictionary of optimized unit cells. The generation of the dictionary is carried out by exploiting micro-SIMP with AnisoTropic mesh adaptivitY (microSIMPATY) algorithm, which promotes the design of free-form layouts. The proposed methodology is particularized in a proof-of-concept setting for the design of orthotic devices for the treatment of foot diseases. Different patient-specific settings drive the prototyping of customized insoles, which are numerically verified and successively validated in terms of mechanical performances and manufacturability.

Staggered DG Method with Small Edges for Darcy Flows in Fractured Porous Media

In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectively. Our current analysis works on fairly general polygonal elements even in the presence of small edges. We prove the optimal convergence estimates in $$L^2$$ error for all the variables by exploiting the Ritz projection. Importantly, our error estimates are shown to be fully robust with respect to the heterogeneity and anisotropy of the permeability coefficients. Several numerical experiments including meshes with small edges and anisotropic meshes are carried out to confirm the theoretical findings. Finally, our method is applied in the framework of unfitted mesh.

High‐order accurate conjugate heat transfer solutions with a finite volume method in anisotropic meshes with application in polymer processing

AbstractThe computational modeling has become an indispensable tool to support the engineering design and control of many industrial processes. In polymer processing applications, the simulation of conjugate heat transfer phenomena is critical as rigorous temperature control is necessary to ensure that the produced parts meet the required specifications. In this article, a high‐order accurate finite volume method is proposed to improve the numerical accuracy and the computational efficiency of conjugate heat transfer simulations with application in polymer processing. Anisotropic meshes are investigated to significantly reduce the number of unknowns in convection‐dominated problems where the higher temperature variations occur perpendicularly to curved boundaries and interfaces. The reconstruction for off‐site data method based on polynomial reconstructions is employed to fulfill the prescribed boundary and interface conditions solely using polygonal meshes to avoid the limitations of curved mesh approaches. A code verification benchmark based on manufactured solutions proves that the proposed method provides a fourth‐order of convergence and is computationally more cost‐effective than the classical second‐order of convergence. Moreover, meshes with higher aspect ratios improve the calculation efficiency but suffer from a small accuracy penalty due to conditioning deterioration. Comparison with the classical finite volume discretization techniques, widely implemented in commercial and open‐source software packages, is also provided. The applicability and performance of the proposed method are further supported with a practical case study for the sheet extrusion cooling stage.

A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh

A virtual element method (VEM) with the first-order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual [Formula: see text]-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.

An anisotropic h-adaptive strategy for discontinuous Petrov-Galerkin schemes using a continuous mesh model

Certain Petrov-Galerkin schemes deliver inherently stable formulations of variational problems on a given mesh by selecting appropriate pairs of trial and test spaces. These schemes are especially suited for adaptation, due to their inherent ability to yield robust a posteriori error estimates. On the other hand, metric-based continuous mesh models have previously been proposed for schemes using piecewise polynomial approximation spaces. These models aim to build (near) optimal anisotropic meshes with respect to interpolation error models. The main focus of this article is to formulate continuous-mesh error models for the optimal Petrov-Galerkin methodology using the inbuilt a posteriori error estimate rather than a generic interpolation error model. This pairs the ability to produce near optimal anisotropic simplex meshes with a numerical method that in turn produces optimal stability and approximation properties on these meshes. Error models are formulated either with respect to a suitable norm of the numerical error, or with respect to certain admissible target functionals, in the spirit of goal-oriented adaptation. We demonstrate the fidelity of the proposed metric-based mesh adaptation strategy via numerical examples for convection-diffusion problems on triangular meshes.

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

Pressure‐robustness for the Stokes equations on anisotropic meshes

AbstractPressure‐robustness has been widely studied since the conception of the notion and the introduction of the reconstruction approach for classical mixed methods in [1]. Using discretizations capable of yielding velocity solutions that are independent of the pressure approximation has been recognized as essential, and a large number of recent articles attest to this fact, e.g., [2, 3]. Apart from the pressure‐robustness aspect, incompressible flows exhibit anisotropic phenomena in the solutions which can be dealt with by using anisotropic mesh grading. The recent publications [4,5] deal with the combination of both challenges. We briefly revisit the results from [4] and provide an insightful new numerical example.

Combined numerical and experimental approach to determine numerical model of abdominal scaffold

A proper junction of the prosthesis and the abdominal wall is important in successful hernia repair. The number of tacks should be balanced to assure appropriate mesh fixation and not to induce post-operative pain. Numerical simulations help to find this balance. The study is aimed at creating a proper numerical model of a knitted surgical mesh subjected to boundary conditions and load occurring in the abdominal cavity. Continuous, anisotropic constitutive relation is considered to reflect the mesh behaviour. Different sets of material law parameters are determined on the basis of different bi-axial tests setups. Force- and displacement-controlled tests with different ratios are considered. Consequently, some numerical model variants are obtained featuring various reaction distributions in the scaffold fixation points. The proper variant is selected based on comparison of the position of maximal reaction force in the numerical model and in the reference physical model of operated hernia. Force-driven tests have shown anisotropic mesh behaviour, while equibiaxial displacement-driven test has demonstrated reduced anisotropic response. Within seven scenarios of constitutive parameters identification (based on single or combined experimental data), the equibiaxial force-controlled test appeared to produce the most relevant model to follow the prosthesis behaviour under pressure. The position of maximal reaction force in such model is similar to obtained in the physical hernia model. The equibiaxial force-driven test provides most suitable data for Gasser-Ogden-Holzapfel constitutive model identification of a considered surgical mesh to be used to model the mesh under pressure.

Periodic adjoints and anisotropic mesh adaptation in rotating frame for high-fidelity RANS turbomachinery applications

The scope of this paper is to demonstrate the viability and efficiency of metric-based unstructured anisotropic mesh adaptation techniques to turbomachinery applications. The main difficulty in turbomachinery is the periodicity of the domain that must be taken into account in the mesh-adaptive solution process. The periodicity is strongly enforced in the flow solver using ghost entities to minimize the impact on the source code. For the mesh adaptation, the local remeshing is done in two steps. First, the inner domain is remeshed with frozen periodic frontiers, and, second, the periodic surfaces are remeshed after moving geometric entities from one side of the domain to the other. One of the main goal of this work is to demonstrate that mesh-independent certified numerical solutions can be obtained thanks to anisotropic mesh adaptation and that it is possible to run high-fidelity CFD on unstructured adapted meshes composed only of tetrahedra. This paper demonstrates how mesh adaptation, thanks to its automation, is able to generate meshes that are extremely difficult to envision and almost impossible to generate manually, leading to highly accurate numerical solutions. This study considers feature-based error estimate based on the standard multi-scale Lp interpolation error estimate and goal-oriented error estimate using an adjoint state to control the error on turbomachinery output functionals. A description of the flow solver and the adjoint solver is given in this work, as they are very different from what is encountered in the turbomachinery community. We also present all the specific modifications that have been introduced in the adaptive process to deal with periodic simulations used for turbomachinery applications. The periodic mesh adaptation strategy is then tested and validated on the LS89 high pressure axial turbine vane and the NASA Rotor 37 test cases.

Anisotropic Tetrahedral Mesh Adaptation with Improved Metric Alignment and Orthogonality

An enhanced mesh adaptation method aiming at generating elements that align the metric is proposed. Well-proven and robust local operators are tailored to modify an existing mesh to satisfy the underlying metric field while enhancing the metric-alignment and orthogonality of the mesh elements. The metric field defines a target element size, shape, and orientation at each point, and a corresponding transformation from a reference element to the target element. A quality criterion based on this transformation is used for selecting edges for splitting and deciding local mesh connectivity after edge collapse, which helps to improve metric alignment and orthogonality of the adapted mesh. Finally, a workflow is organized by combining the local operations and the new techniques to adapt a mesh according to a metric field. Numerical experiments on anisotropic meshes are presented to demonstrate the effectiveness of the proposed method.

Modeling and simulation of the lateral photovoltage scanning method

The goal of this paper is to model and simulate the lateral photovoltage scanning (LPS) method which detects inhomogeneities in semiconductor crystals. Our drift-diffusion model is based on the semiconductor device equations combined with an implicit boundary condition which models a volt meter. To validate our 2D and 3D finite volume simulations, we use theory developed by Tauc [1] to derive three analytical predictions which our simulation results corroborate, even for anisotropic 2D and 3D meshes. Our code runs about two orders of magnitudes faster than earlier implementations based on commercial software [2]. It also performs well for small doping concentrations which previously could not be simulated at all due to numerical instabilities. We present a convergence study which shows that the LPS voltage converges quadratically. Finally, our simulations provide experimentalists with reference laser powers for which meaningful voltages can still be measured. For higher laser power the screening effect does not allow this anymore.

An efficient edge based data structure for the compressible Reynolds‐averaged Navier–Stokes equations on hybrid unstructured meshes

AbstractAn efficient edge based data structure has been developed in order to implement an unstructured vertex based finite volume algorithm for the Reynolds‐averaged Navier–Stokes equations on hybrid meshes. In the present approach, the data structure is tailored to meet the requirements of the vertex based algorithm by considering data access patterns and cache efficiency. The required data are packed and allocated in a way that they are close to each other in the physical memory. Therefore, the proposed data structure increases cache performance and improves computation time. As a result, the explicit flow solver indicates a significant speed up compared to other open‐source solvers in terms of CPU time. A fully implicit version has also been implemented based on the PETSc library in order to improve the robustness of the algorithm. The resulting algebraic equations due to the compressible Navier–Stokes and the one equation Spalart–Allmaras turbulence equations are solved in a monolithic manner using the restricted additive Schwarz preconditioner combined with the FGMRES Krylov subspace algorithm. In order to further improve the computational accuracy, the multiscale metric based anisotropic mesh refinement library PyAMG is used for mesh adaptation. The numerical algorithm is validated for the classical benchmark problems such as the transonic turbulent flow around a supercritical RAE2822 airfoil and DLR‐F6 wing‐body‐nacelle‐pylon configuration. The efficiency of the data structure is demonstrated by achieving up to an order of magnitude speed up in CPU times.

Demonstration of a three-dimensional dynamically adaptive atmospheric dynamic framework for the simulation of mountain waves

In this paper, Fluidity-Atmosphere, representative of a three-dimensional (3D) non-hydrostatic Galerkin compressible atmospheric dynamic framework, is generated to resolve large-scale and small-scale phenomena simultaneously. This achievement is facilitated by the use of non-hydrostatic equations and the adoption of a flexible 3D dynamically adaptive mesh where the mesh is denser in areas with higher gradients of variable solutions and relatively sparser in the rest of the domain while maintaining promising accuracy and reducing computational resource requirements. The dynamic core is formulated based on anisotropic tetrahedral meshes in both the horizontal and vertical directions. The performance of the adaptive mesh techniques in Fluidity-Atmosphere is evaluated by simulating the formation and propagation of a non-hydrostatic mountain wave. The 2D anisotropic adaptive mesh shows that the numerical solution is in good agreement with the analytic solution. The variation in the horizontal and vertical resolutions has a strong impact on the smoothness of the results and maintains convergence even at high resolutions. When the simulation is extended to 3D, Fluidity-Atmosphere shows stable and symmetric results in the benchmark test cases. The flows over a bell-shaped mountain are resolved quite smoothly. For steep mountains, Fluidity-Atmosphere performs very well, which shows the potential of using 3D adaptive meshes in atmospheric modeling. Finally, as an alternative cut-cell mesh in Fluidity-Atmosphere, the anisotropic adaptive mesh coupled with the Galerkin method provides an alternative accurate representation of terrain-induced flow.

A higher-order finite element method with unstructured anisotropic mesh adaption for two phase flows with surface tension

A novel finite element framework is proposed for the numerical simulation of two phase flows with surface tension. The Level-Set (LS) method with piece-wise quadratic (P2) interpolation for the liquid–gas interface is used in order to reach higher-order convergence rates in regions with smooth interface. A balanced-force implementation of the continuum surface force model is used to take into account the surface tension and to solve static problems as accurately as possible. Given that this requires a balance between the discretization used for the LS function, and that used for the pressure field, an equal-order P2/P2/P2 scheme is proposed for the Navier–Stokes and LS advection equations, which are strongly coupled with each other. This fully implicit formulation is stabilized using the residual-based variational multiscale framework. In order to improve the accuracy and obtain optimal convergence rates with a minimum number of elements, an anisotropic mesh adaption method is proposed where the unstructured mesh is kept as fine as possible close to the zero iso-value of the P2 LS function. Elements are automatically stretched in regions with flat interface in order to keep the complexity fixed during the simulation. The accuracy and efficiency of this approach are demonstrated for two and three dimensional simulations of a rising bubble.

Parametric Study of Nonequilibrium Shock Interference Patterns over a Fuselage-and-Wing Conceptual Vehicle

Predicting shock/shock and shock/boundary-layer interactions in gas flows that surround high-speed vehicles is key in aerodynamic design. Under typical hypersonic conditions, these flow structures are influenced by complex nonequilibrium phenomena leading to high-temperature effects. In this work, the conceptual Bedford wing-body vehicle is studied to analyze flow patterns in shock/shock and shock/boundary-layer interactions with thermochemical nonequilibrium. A parametric computational fluid dynamics study is carried out for different hypersonic operating conditions, with respect to the freestream Mach number. Simulations are performed with the SU2-NEMO solver coupled to the Mutation++ library, which provides all the necessary thermodynamic, kinetic, and transport properties of the mixture and chemical species. The Adaptive Mesh Generation library is used for automatic anisotropic mesh adaptation. Numerical results show that increasing the freestream Mach number from 4 to 10 leads to changes in the shock layer, locations of shock impingement, and regions of boundary-layer separation. Despite these changes, the change in freestream Mach number has little impact on the overall shock interaction structures.