Higher-order displacement-based finite element methods are useful for simulating bending problems and potentially addressing mesh-locking associated with nearly-incompressible elasticity,yet are computationally expens...Higher-order displacement-based finite element methods are useful for simulating bending problems and potentially addressing mesh-locking associated with nearly-incompressible elasticity,yet are computationally expensive.To address the computational expense,the paper presents a matrix-free,displacement-based,higher-order,hexahedral finite element implementation of compressible and nearly-compressible(ν→0.5)linear isotropic elasticity at small strain with p-multigrid preconditioning.The cost,solve time,and scalability of the implementation with respect to strain energy error are investigated for polynomial order p=1,2,3,4 for compressible elasticity,and p=2,3,4 for nearly-incompressible elasticity,on different number of CPU cores for a tube bending problem.In the context of this matrix-free implementation,higher-order polynomials(p=3,4)generally are faster in achieving better accuracy in the solution than lower-order polynomials(p=1,2).However,for a beam bending simulation with stress concentration(singularity),it is demonstrated that higher-order finite elements do not improve the spatial order of convergence,even though accuracy is improved.展开更多
In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian(ALE)one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimension...In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian(ALE)one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes.A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in[25].For that purpose,a new element-local weak formulation of the governing PDE is adopted on moving space-time elements.The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.Moreover,a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element.To maintain algorithmic simplicity,the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges.This is possible in the ALE framework,which allows a local mesh velocity that is different from the local fluid velocity.We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.展开更多
In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The highorder variational reconstructio...In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The highorder variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4 est based on the forest of octrees is adopted. Several two-and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.展开更多
In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and com...In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and computationally efficient Finite Volume(FV)scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions,respectively.The Multi-dimensional Optimal Order Detection(MOOD)method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes.It is an arbitrary high-order accurate Finite Volume scheme in space,using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.In the following work,the time discretization is performed with an elegant and efficient one-step ADER procedure.Doing so,we retain the good properties of the MOOD scheme,that is to say the optimal high-order of accuracy is reached on smooth solutions,while spurious oscillations near singularities are prevented.The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D,but it also increases the stability of the overall scheme.A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost,robustness,accuracy and efficiency.The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive(memory and/or CPU time),or because it is more accurate for a given grid resolution.A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws:the Euler equations of compressible gas dynamics,the classical equations of ideal magneto-Hydrodynamics(MHD)and finally the relativistic MHD equations(RMHD),which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.All tests are run on genuinely unstructured grids composed of simplex elements.展开更多
A second nonoscillatory finite volume scheme, based on a new linear reconstruction to the flow variable, is given for 2d hyperbolic conservation laws on unstructured triangular mesh. Numerical results for plate shock ...A second nonoscillatory finite volume scheme, based on a new linear reconstruction to the flow variable, is given for 2d hyperbolic conservation laws on unstructured triangular mesh. Numerical results for plate shock wave reflection and cavity flow are presented.展开更多
基金The research relied on computational resources[29]provided by the University of Colorado Boulder Research Computing Group,which is supported by the National1302 CMES,2021,vol.129,no.3 Science Foundation(Awards ACI-1532235 and ACI-1532236)University of Colorado Boulder,and Colorado State University.
文摘Higher-order displacement-based finite element methods are useful for simulating bending problems and potentially addressing mesh-locking associated with nearly-incompressible elasticity,yet are computationally expensive.To address the computational expense,the paper presents a matrix-free,displacement-based,higher-order,hexahedral finite element implementation of compressible and nearly-compressible(ν→0.5)linear isotropic elasticity at small strain with p-multigrid preconditioning.The cost,solve time,and scalability of the implementation with respect to strain energy error are investigated for polynomial order p=1,2,3,4 for compressible elasticity,and p=2,3,4 for nearly-incompressible elasticity,on different number of CPU cores for a tube bending problem.In the context of this matrix-free implementation,higher-order polynomials(p=3,4)generally are faster in achieving better accuracy in the solution than lower-order polynomials(p=1,2).However,for a beam bending simulation with stress concentration(singularity),it is demonstrated that higher-order finite elements do not improve the spatial order of convergence,even though accuracy is improved.
基金the European Research Council(ERC)under the European Union’s Seventh Framework Programme(FP7/2007-2013)with the research project STiMulUs,ERC Grant agreement no.278267.
文摘In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian(ALE)one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes.A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in[25].For that purpose,a new element-local weak formulation of the governing PDE is adopted on moving space-time elements.The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.Moreover,a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element.To maintain algorithmic simplicity,the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges.This is possible in the ALE framework,which allows a local mesh velocity that is different from the local fluid velocity.We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.
基金supported by the National Natural Science Foundation of China(Nos.91752114 and 11672160)
文摘In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The highorder variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4 est based on the forest of octrees is adopted. Several two-and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.
基金the European Research Council(ERC)under the European Union’s Seventh Framework Programme(FP7/2007-2013)the research project STiMulUs,ERC Grant agreement no.278267+1 种基金.R.L.has been partially funded by the ANR under the JCJC project“ALE INC(ubator)3D”the reference LA-UR-13-28795.The authors would like to acknowledge PRACE for awarding access to the SuperMUC supercomputer based in Munich,Germany at the Leibniz Rechenzentrum(LRZ)。
文摘In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and computationally efficient Finite Volume(FV)scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions,respectively.The Multi-dimensional Optimal Order Detection(MOOD)method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes.It is an arbitrary high-order accurate Finite Volume scheme in space,using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.In the following work,the time discretization is performed with an elegant and efficient one-step ADER procedure.Doing so,we retain the good properties of the MOOD scheme,that is to say the optimal high-order of accuracy is reached on smooth solutions,while spurious oscillations near singularities are prevented.The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D,but it also increases the stability of the overall scheme.A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost,robustness,accuracy and efficiency.The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive(memory and/or CPU time),or because it is more accurate for a given grid resolution.A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws:the Euler equations of compressible gas dynamics,the classical equations of ideal magneto-Hydrodynamics(MHD)and finally the relativistic MHD equations(RMHD),which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.All tests are run on genuinely unstructured grids composed of simplex elements.
文摘A second nonoscillatory finite volume scheme, based on a new linear reconstruction to the flow variable, is given for 2d hyperbolic conservation laws on unstructured triangular mesh. Numerical results for plate shock wave reflection and cavity flow are presented.
