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This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in if(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation--histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675-A702]), efficient matrix-free preconditioners can be constructed for the (1,1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. [ABSTRACT FROM AUTHOR]

A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is here extended to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms.

In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H (curl) and H (div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated. [ABSTRACT FROM AUTHOR]

This work compares two Nitsche‐type approaches to treat non‐conforming triangulations for a high‐order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point‐to‐point interpolation) uses inconsistent integration with quadrature points prescribed by a primary element. The second approach uses consistent integration by choosing quadratures depending on the intersection between non‐conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point‐to‐point interpolation. However, we show that this approach can not safely be used for DG discretizations of the acoustic conservation equations since, in our setting, it yields spurious oscillations that lead to instabilities. This work presents a test case in that we can observe the instabilities and shows that consistent integration is required to maintain a stable method. Additionally, we provide a detailed analysis of the method with consistent integration. We show optimal spatial convergence rates globally and in each mesh region separately. The method is constructed such that it can natively treat overlaps between elements. Finally, we highlight the benefits of non‐conforming discretizations in acoustic computations by a numerical test case with different fluids. [ABSTRACT FROM AUTHOR]

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the discrete ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory (LLNL). Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations. [ABSTRACT FROM AUTHOR]

Based on the Carrera unified formulation (CUF) and first-invariant hyperelasticity, this work proposes a displacement-based high order one-dimensional (1 D) finite element model for the geometrical and physical nonlinear analysis of isotropic, slightly compressible soft material structures. Different strain energy functions are considered and they are decomposed in a volumetric and an isochoric part, the former acting as penalization of incompressibility. Given the material Jacobian tensor, the nonlinear governing equations are derived in a unified, total Lagrangian form by expanding the three-dimensional displacement field with arbitrary cross-section polynomials and using the virtual work principle. The exact analytical expressions of the elemental internal force vector and tangent matrix of the unified beam model are also provided. Several problems are addressed, including uniaxial tension, bending of a slender structure, compression of a three-dimensional block, and a thick pinched cylinder. The proposed model is compared with analytical solutions and literature results whenever possible. It is demonstrated that, although 1 D, the present CUF-based finite element can address simple to complex nonlinear hyperelastic phenomena, depending on the theory approximation order. [ABSTRACT FROM AUTHOR]

In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation. [ABSTRACT FROM AUTHOR]

When a numerical method is used to analyze or design an electromagnetic structure, as the finite element method, the introduction of the behavior of some component of the electromagnetic field (when this is known) allows to project the physical problem onto a two‐dimensional mesh, multiplying the capacity of the method to make efficient computer aided design of complex devices as they are demanded nowadays. Though this is an old issue, which has been focused one or two decades ago, the arrival of higher‐order function basis and powerful tools of meshing and post processing makes interesting to review this important field within the computational electromagnetic area. [ABSTRACT FROM AUTHOR]

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations. [ABSTRACT FROM AUTHOR]

In this study, we present the bicubic Hermite element method (BHEM), a new computational framework devised for the elastodynamic simulation of thin-shell structures. The BHEM is constructed based on quadrilateral Hermite patches, which serve as a unified representation for shell geometry, simulation, collision avoidance, as well as rendering. Compared with the commonly utilized linear FEM, the BHEM offers higher-order solution spaces, enabling the capture of more intricate and smoother geometries while employing significantly fewer finite elements. In comparison to other high-order methods, the BHEM achieves conforming C 1 continuity for Kirchhoff–Love (KL) shells with minimal complexity. Furthermore, by leveraging the subdivision and convex hull properties of Hermite patches, we develop an efficient algorithm for ray-patch intersections, facilitating collision handling in simulations and ray tracing in rendering. This eliminates the need for laborious remodeling of the pre-existing surface as the conventional approaches do. We substantiate our claims with comprehensive experiments, which demonstrate the high accuracy and versatility of the proposed method. [Display omitted] • The BHEM serves as a unified representation for Kirchhoff–Love shells. • The high-order solution spaces offered by BHEM naturally enables G1 continuity. • An efficient algorithm for ray-patch intersections is developed for BHEM. • Comprehensive experiments demonstrate BHEM's accuracy and versatility. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a nonoverlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Pad\'e approximants of the nonlocal magnetic-to-electric operators. Numerical results are presented to validate and analyze the new approach for both homogeneous and inhomogeneous scatterers. [ABSTRACT FROM AUTHOR]

We propose a new Nitsche-type approach for weak enforcement of normal velocity boundary conditions for a Lagrangian discretization of the compressible shock-hydrodynamics equations using high-order finite elements on curved boundaries. Specifically, the variational formulation is appropriately modified to enforce free-slip wall boundary conditions, without perturbing the structure of the function spaces used to represent the solution, with a considerable simplification with respect to traditional approaches. Total energy is conserved and the resulting mass matrices are constant in time. The robustness and accuracy of the proposed method are validated with an extensive set of tests involving nontrivial curved boundaries. [ABSTRACT FROM AUTHOR]

• The optimal convergence rate of L 2 -error for the velocity is completely proven. • The corresponding algorithm of our method is simple to be implemented by codes. • The remarkable advantage of our method is demonstrated via adequate numerical experiments. In this work, we develop a high-order pressure-robust method for the rotation form of the incompressible Navier–Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H (div) -conforming velocity reconstruction operator. In order to match the rotation form and ease error analysis, a skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proved to achieve the pressure-independent velocity errors. Optimal convergence rates of H 1 , L 2 -error for the velocity and L 2 -error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and remarkable performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we introduce general transfer operators between high-order and loworder refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving, and high-order accurate. While the proof holds for affine geometries, numerical experiments indicate that the results hold for more general curved meshes. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multidiscretization coupling. [ABSTRACT FROM AUTHOR]

As the use of spectral/hp element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hp element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hp element library Nektar++ by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst 50 % slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of 7 × . Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to 30 % speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around 6 × speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires O (k) storage compared to a best case space complexity of O (k 2) for the Lagrangian interpolation matrix method. [ABSTRACT FROM AUTHOR]

The moving discontinuous Galerkin finite element method with interface condition enforcement is applied to the case of viscous flows. This method uses a weak formulation that separately enforces the conservation law, constitutive law, and the corresponding interface conditions in order to provide the means to detect interfaces or underresolved flow features. To satisfy the resulting overdetermined weak formulation, the discrete domain geometry is introduced as a variable, so that the method implicitly fits a priori unknown interfaces and moves the grid to resolve sharp, but smooth, gradients, achieving a form of anisotropic curvilinear r‐adaptivity. This approach avoids introducing low‐order errors that arise using shock capturing, artificial dissipation, or limiting. The utility of this approach is demonstrated with its application to a series of test problems culminating with the compressible Navier–Stokes solution to a Mach 5 viscous bow shock for a Reynolds number of 105 in two‐dimensional space. Time accurate solutions of unsteady problems are obtained via a space‐time formulation, in which the unsteady problem is formulated as a higher dimensional steady space‐time problem. The method is shown to accurately resolve and transport viscous structures without relying on numerical dissipation for stabilization. [ABSTRACT FROM AUTHOR]

With the gap between processor clock speeds and memory bandwidth speeds continuing to increase, the use of arithmetically intense schemes, such as high-order finite element methods, continues to be of considerable interest. In particular, the use of matrix-free formulations of finite element operators for tensor-product elements of quadrilaterals in two dimensions and hexahedra in three dimensions, in combination with single-instruction multiple-data instruction sets, is a wellstudied topic at present for the efficient implicit solution of elliptic equations. However, a considerable limiting factor for this approach is the use of meshes comprising of only quadrilaterals or hexahedra, the creation of which is still an open problem within the mesh generation community. In this article, we study the efficiency of high-order finite element operators for the Helmholtz equation with a focus on extending this approach to unstructured meshes of triangles, tetrahedra, and prismatic elements using the spectral/hp element method and corresponding tensor-product bases for these element types. We show that although performance is naturally degraded when going from hexahedra to these simplicial elements, efficient implementations can still be obtained that are capable of attaining 50% through 70% floating point operations of the peak of processors with both AVX2 and AVX512 instruction sets. [ABSTRACT FROM AUTHOR]

Summary: The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the underresolved regime, mass conservation and energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high‐order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard L2‐based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence‐free H(div)‐conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. This work raises the question whether and to which extent these two approaches are equivalent when applied to underresolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for underresolved simulations of turbulent flows due to their inherent dissipation mechanisms. [ABSTRACT FROM AUTHOR]

We assess the ability of three different approaches based on high-order discontinuous Galerkin methods to simulate under-resolved turbulent flows. The capabilities of the mass conserving mixed stress method as structure resolving large eddy simulation solver are examined. A comparison of a variational multiscale model to no-model or an implicit model approach is presented via numerical results. In addition, we present a novel approach for turbulent modeling in wall-bounded flows. This new technique provides a more accurate representation of the actual subgrid scales in the near wall region and gives promising results for highly under-resolved flow problems. In this paper, the turbulent channel flow and periodic hill flow problem are considered as benchmarks for our simulations. • Low dissipative high-order DG-FEM solver for implicit large eddy simulation. • Novel VMS like upwind stabilization for incompressible wall-bounded turbulent flows. • Comparison and verification using different high Reynolds number test cases. [ABSTRACT FROM AUTHOR]

Summary: The need to compute the intersections between a line and a high‐order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a noniterative method for computing intersections by solving a matrix singular value decomposition and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one‐shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton‐Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree q of the shape functions and is 2q × 3q for curves and 6q2 × 8q2 for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high‐order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/. [ABSTRACT FROM AUTHOR]

The aim of this work is to propose a novel, fast solver for the Poisson problem discretized with high-order spectral element methods (HO-SEM) in canonical geometries (rectangle in two dimensions, rectangular parallelepiped in three dimensions). This method is based on the use of the discrete Fourier transform to reduce the problem to the inversion of the symbol of the operator in the frequency space. The solver proposed is endowed with several properties. First, it preserves the efficiency of the standard FFT algorithm; then, the matrix storage is drastically reduced (in particular, it is independent of the space dimension); a pseudoexplicit singular value decomposition is used for the inversion of the symbols; and finally, it can be extended to nonperiodic boundary conditions. Furthermore, due to the underlying HO-SEM discretization, the multidimensional symbol of the operator can be efficiently computed from the one-dimensional symbol by tensorisation. [ABSTRACT FROM AUTHOR]

The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The discrete crack limit case is approached for a small length-scale parameter that controls the width of the transition region between the fully broken and the undamaged phases. From a computational standpoint, this necessitates fine meshes, at least locally, in order to accurately resolve the phase-field profile. In the classical approach, phase-field models are computed on a fixed mesh that is a priori refined in the areas where the crack is expected to propagate. This on the other hand curbs the convenience of using phase-field models for unknown crack paths and its ability to handle complex crack propagation patterns. In this work, we overcome this issue by employing the multi-level hp-refinement technique that enables a dynamically changing mesh which in turn allows the refinement to remain local at singularities and high gradients without problems of hanging nodes. Yet, in case of complex geometries, mesh generation and in particular local refinement becomes non-trivial. We address this issue by integrating a two-dimensional phase-field framework for brittle fracture with the finite cell method (FCM). The FCM based on high-order finite elements is a non-geometry-conforming discretization technique wherein the physical domain is embedded into a larger fictitious domain of simple geometry that can be easily discretized. This facilitates mesh generation for complex geometries and supports local refinement. Numerical examples including a comparison to a validation experiment illustrate the applicability of the multi-level hp-refinement and the FCM in the context of phase-field simulations. [ABSTRACT FROM AUTHOR]

We spatially discretize the discrete ordinates radiation transport equation using high-order discontinuous Galerkin finite elements in R-Z geometry. Previous research has demonstrated first-order methods have 2nd-order spatial convergence rates in R-Z geometry. Presently, we demonstrate that higher-order (HO) methods preserve the p + 1 convergence rates on smooth solutions, where p is the finite element order. Further, we extend the use of HO finite element methods to utilize meshes with curved surfaces. We also demonstrate that meshes with curved surfaces do not degrade the observed spatial convergence rates. Finally, we exercise the methodology on a highly diffusive and scattering problem with alternating incident boundaries to show that both HO methods and mesh refinement reduce the negative scalar fluxes that result from oscillations. [ABSTRACT FROM AUTHOR]

Summary: A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high‐order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space‐time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three‐dimensional flows that provide an initial assessment of the stability and accuracy of the method. A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The method preserves a high‐order representation up to the interface, without relying on artificial dissipation or shock capturing for stabilization. The method can compute curved interfaces, with a priori unknown topology, in multiple dimensions. The method applies to both steady flows and unsteady flows via a space‐time formulation. [ABSTRACT FROM AUTHOR]

We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of [P. Knupp, Eng. Comput., 28 (2012), pp. 419--429]. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Lagrangian--Eulerian (ALE) application [BLAST: High-order curvilinear finite elements for shock hydrodynamics. [ABSTRACT FROM AUTHOR]

In order to reduce the transfer of sound and vibrations in structures such as timber buildings, thin elastomer layers can be embedded between their components. The influence of these elastomers on the response of the structures in the low frequency range can be determined accurately by using conforming hexahedral finite elements. Three-dimensional mesh generation, however, is yet a non-trivial task and mesh refinements which may be necessary at the junctions can cause a high computational effort. One remedy is to mesh the components independently from each other and to couple them using the mortar method. Further, the hexahedral mesh for the thin elastomer layer itself can be avoided by integrating its elastic behavior into the mortar formulation. The present paper extends this mortar formulation to take damping into account such that frequency response analyses can be performed more accurately. Finally, the proposed method is verified by numerical examples. [ABSTRACT FROM AUTHOR]

We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit time stepping is used for the temporal discretization, which allows for a much larger time step size for stability compared to explicit methods, especially for low-Mach number flows and/or on highly distorted meshes. Ample numerical results are presented to showcase the good performance of our proposed scheme. • A Lagrangian scheme based on Energetic Variational Approach for compressible flows. • Mass, momentum, energy conservation and entropy stability for spatial discretization. • Fully implicit time stepping that allows large time step size. • High order space/time accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes a high-order two-dimensional (2D) finite element model for the analysis of isotropic, nearly incompressible hyperelastic material structures based on a decoupled Neo–Hookean strain energy function. The model is based on the Carrera Unified Formulation (CUF) , which allows to automatically implement different kinematics by using an opportune recursive notation. The principle of virtual work and a finite element approximation are exploited to obtain the nonlinear governing equations. Considering the three-dimensional full Green–Lagrange strain components and given the material Jacobian tensor, the explicit forms of tangent stiffness matrices of unified plate elements are presented in terms of the fundamental nuclei, which are independent of the theory approximation order. Several problems of soft material plates under uniform pressure are investigated, including a silicone rubber clamped plate and a simply supported plate made of biological material. The proposed model is compared with literature results including those coming from experiments and numerical solutions. The numerical investigation demonstrated the validity and accuracy of the proposed methodology for the analysis of hyperelastic plates. • Unified plate finite elements with scalable accuracy are developed. • The model accounts for full Green–Lagrange strains. • It is able top study large strains of thin- top thick plates made of hyperelastic materials. • The results are verified against literature results and experiments. [ABSTRACT FROM AUTHOR]

We present a new perspective on the use of weighted essentially nonoscillatory (WENO) reconstructions in high-order methods for scalar hyperbolic conservation laws. The main focus of this work is on nonlinear stabilization of continuous Galerkin (CG) approximations. The proposed methodology also provides an interesting alternative to WENO-based limiters for discontinuous Galerkin (DG) methods. Unlike Runge–Kutta DG schemes that overwrite finite element solutions with WENO reconstructions, our approach uses a reconstruction-based smoothness sensor to blend the numerical viscosity operators of high- and low-order stabilization terms. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while preserving the high-order accuracy of a linearly stable baseline discretization in regions where the exact solution is sufficiently smooth. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive error estimates and perform numerical experiments. In particular, we prove that the consistency error of the nonlinear stabilization is of the order p + 1 / 2 , where p is the polynomial degree. This estimate is optimal for general meshes. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as p + 1. • A new kind of WENO stabilization for finite element methods. • Applicability to high-order continuous Galerkin approximations. • A smoothness sensor that uses Hermite polynomial reconstructions. • Adaptive blending of high- and low-order stabilization operators. • A priori analysis of accuracy and consistency for smooth data. [ABSTRACT FROM AUTHOR]

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin (HDG) method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the HDG method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. It is found that high-order continuous finite elements give best time to solution for smooth solutions, closely followed by the matrix-free solvers for the other two discretizations. Their performance is up to an order of magnitude higher than that of the best matrix-based methods, even after including the superconvergence effects in the matrix-based HDG method. This difference is because of the vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A rooine performance model confirms the superiority of the matrix-free implementation. [ABSTRACT FROM AUTHOR]

Abstract: This paper combines the use of high‐order finite element methods with parallel preconditioners of domain decomposition type for solving electromagnetic problems arising from brain microwave imaging. The numerical algorithms involved in such complex imaging systems are computationally expensive since they require solving the direct problem of Maxwell equations several times. Moreover, wave propagation problems in the high‐frequency regime are challenging because a sufficiently high number of unknowns are required to accurately represent the solution. To use these algorithms in practice for brain stroke diagnosis, running time should be reasonable. The method presented in this paper, coupling high‐order finite elements and parallel preconditioners, makes it possible to reduce the overall computational cost and simulation time while maintaining accuracy. [ABSTRACT FROM AUTHOR]

A convergence study of the forward problem of electrical impedance tomography is performed using triangular high-order piecewise polynomial finite-element methods (p-FEM) on a square domain. The computation ofp-FEM for the complete electrode model (CEM) is outlined and a novel analytic solution to the CEM on a square domain is presented. Errors as a function of mesh-refinement and computational time, as well as convergence rates as a function of contact impedance, are computed numerically for different polynomial approximation orders. It is demonstrated thatp-FEM can generate more accurate forward solutions in less computational time, which implies more accurate simulated interior potentials, electrode voltages and conductivity Jacobians. [ABSTRACT FROM AUTHOR]

A novel numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF) in which the geometry is represented by orthonormal vectors, the proposed scheme not only has the fewest dimensionality both in space and time, but also does not require either of metric tensors, composite meshes, or the ambient space. The MMF–SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order p in space and an explicit Runge–Kutta scheme in time. The numerical model is validated against six standard tests on the sphere and the optimal order of convergence of p + 1 is numerically demonstrated. The MMF–SWE scheme is also demonstrated for its efficiency and stability on the general rotating surfaces such as ellipsoid, irregular, and non-convex surfaces. [ABSTRACT FROM AUTHOR]

We present three new sets of C1 hierarchical high-order tensor-product bases for conforming finite elements. The first basis is a high-order extension of the Bogner-Fox-Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C1 global continuity on the common edges of elements. The second basis uses the tensor product of fifth-order Hermite polynomials and high-order functions and achieves global C1 continuity for meshes of quadrilaterals and C2 continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one-dimensional functions defined in barycentric coordinates. It also has global C1 continuity on edges and C2 continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h- and p-refinements are evaluated by comparing the approximation errors in the L2- and energy norms. A plate with singularity is then studied, and h- and p-refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non-distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so-called non-uniform rational B-spline (NURBS)-enhanced finite element method (NEFEM). The meshes generated with the proposed approach account for the computer-aided design boundary representation of the domain given by NURBS curves. The characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features. The proposed strategy allows to circumvent the time-consuming process of de-featuring complex geometric models before a finite element mesh suitable for the analysis can be produced. A generalisation of the original definition of a NEFEM element is also proposed, enabling to treat more complicated elements with an edge defined by several NURBS curves or more than one edge defined by different NURBS. Three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators. Finally, a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis. ©2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive framework for the simulation of Multifluid flows based on the implicit level-set representation of interfaces and on an efficient solving strategy of the Navier-Stokes equations. The mathematical framework relies on a modular coupling approach between the level-set advection and the fluid equations. The space discretization is performed with possibly high-order stable finite elements while the time discretization features implicit Backward Differentation Formulae of arbitrary order. This framework has been implemented within the Feel++ library, and features seamless distributed parallelism with fast assembly procedures for the algebraic systems and efficient preconditioning strategies for their resolution. We also present simulation results for a three-dimensional Multifluid benchmark, and highlight the importance of using high-order finite elements for the level-set discretization for problems involving the geometry of the interfaces, such as the curvature or its derivatives.

We present an efficient algorithmic framework for constructing multi-level h p -bases that uses a data-oriented approach that easily extends to any number of dimensions and provides a natural framework for performance-optimized implementations. We only operate on the bounding faces of finite elements without considering their lower-dimensional topological features and demonstrate the potential of the presented methods using a newly written open-source library. First, we analyze a Fichera corner and show that the framework does not increase runtime and memory consumption when compared against the classical p -version of the finite element method. Then, we compute a transient example with dynamic refinement and derefinement, where we also obtain the expected convergence rates and excellent performance in computing time and memory usage. • The multi-level hp -method is a simple framework for constructing hp -bases. • Hanging nodes are omitted by supporting basis functions on parent cells. • Our data-oriented extension simplifies applications in higher dimensions. • We obtain exponential convergence with polynomial runtime and memory consumption. • Transient refinements and derefinements are possible using Rothe's method. [ABSTRACT FROM AUTHOR]

As computing hardware evolves, increasing core counts mean that memory bandwidth is becoming the deciding factor in attaining peak performance of numerical methods. High-order finite element methods, such as those implemented in the spectral/ h p framework Nektar++ , are particularly well-suited to this environment. Unlike low-order methods that typically utilise sparse storage, matrices representing high-order operators have greater density and richer structure. In this paper, we show how these qualities can be exploited to increase runtime performance on nodes that comprise a typical high-performance computing system, by amalgamating the action of key operators on multiple elements into a single, memory-efficient block. We investigate different strategies for achieving optimal performance across a range of polynomial orders and element types. As these strategies all depend on external factors such as BLAS implementation and the geometry of interest, we present a technique for automatically selecting the most efficient strategy at runtime. [ABSTRACT FROM AUTHOR]

Finite elements of degree two or more are needed to solve various P.D.E. problems. This paper discusses a method to validate such meshes for the case of the usual Lagrange elements of various degrees. The first section of this paper comes back to Bézier curve and Bézier patches of arbitrary degree. The way in which a Bézier patch and a finite element are related is recalled. The usual Lagrange finite elements of various degrees are discussed, including simplices (triangle and tetrahedron), quads, prisms (pentahedron), pyramids and hexes together with some low-degree Serendipity elements. A validity condition, the positivity of the jacobian, is exhibited for these elements. Elements of various degrees are envisaged also including some 'linear' elements (therefore straight-sided elements of degree 1) because the jacobian (polynomial) of some of them is not totally trivial. [ABSTRACT FROM AUTHOR]

Motivated by the need for efficient and accurate simulation of the dynamics of the polar ice sheets, we design high-order finite element discretizations and scalable solvers for the solution of nonlinear incompressible Stokes equations. In particular, we focus on power-law, shear thinning rheologies commonly used in modeling ice dynamics and other geophysical flows. We use nonconforming hexahedral meshes and the conforming inf-sup stable finite element velocity-pressure pairings Qk × Qk-2disc or Qk × Pk-1disc, where k ≥ 2 is the polynomial order of the velocity space. To solve the nonlinear equations, we propose a Newton-Krylov method with a block upper triangular preconditioner for the linearized Stokes systems. The diagonal blocks of this preconditioner are sparse approximations of the (1,1)-block and of its Schur complement. The (1,1)-block is approximated using linear finite elements based on the nodes of the high-order discretization, and the application of its inverse is approximated using algebraic multigrid with an incomplete factorization smoother. This preconditioner is designed to be efficient on anisotropic meshes, which are necessary to match the high aspect ratio domains typical for ice sheets. As part of this work, we develop and make available extensions to two libraries--a hybrid meshing scheme for the p4est parallel adaptive mesh refinement library and a modified smoothed aggregation scheme for PETSc--to improve their support for solving PDEs in high aspect ratio domains. In a comprehensive numerical study, we find that our solver yields fast convergence that is independent of the element aspect ratio, the occurrence of nonconforming interfaces, and the mesh refinement and that depends only weakly on the polynomial finite element order. We simulate the ice flow in a realistic description of the Antarctic ice sheet derived from field data and study the parallel scalability of our solver for problems with up to 383 million unknowns. [ABSTRACT FROM AUTHOR]

In this article the multiplicative decomposition of the deformation gradient into one part constrained in the direction of the axis of anisotropy and one part describing the directional deformation is proposed. This leads to a clear division of the deformation and stress states in the direction of anisotropy and a remaining part. The decomposition is explained in detail and a constitutive model of hyper-elasticity is proposed for the case of transversal isotropy, where the behavior of the model is investigated with the aid of simple analytical examples. The model is also investigated for inhomogeneous deformation states using high-order finite elements based on hierarchical shape functions showing the sensitivity of the accuracy of the results in the case of anisotropic media. [ABSTRACT FROM AUTHOR]

Highlights: [•] Overview of the current state of the art of high-order fictitious domain methods. [•] Review of the recently introduced Finite Cell Method. [•] Introduces FCMLab as an entrypoint into high-order fictitious domain methods. [•] Information provided allows researchers to quickly prototype new numerical ideas. [Copyright &y& Elsevier]

Abstract: We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/hp elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the computational cost of anatomically-realistic models can be significantly reduced through the surface representation and use of high-order methods. We describe and validate our discretisation and provide a demonstration of its application to modelling electrochemical propagation across a human left atrium. [Copyright &y& Elsevier]