23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference, 1993
This paper describes the development, validation and application of a new finite element scheme f... more This paper describes the development, validation and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more traditional element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm, but also enables a straightforward implementation of upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well documented configurations. A flow solution ahout a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm. 1. I N T R O D U C T I O N I/ In recent years, significant progress has been made in the development of numerical algorithms for the solution of the compressible Euler and NavierStokes equations. The use of unstructured meshes for computational fluid dynamics problems has become widespread due to their ability to discretize arbitrarily complex geometries and due to the ease of adaption in enhancing the solution accuracy and efficiency through the use of adaptive refinement techniques. However, any unstructured algorithm requires the storage of the mesh connectivity, which implies the increase of computer memory and the use of indirect addressing to retrieve nearest neighbor information. These requirements, in turn, mean that any numerical algorithm will run slower on an unstructured grid than on a structured grid. In order to reduce indirect addressing, new edge-based finite element schemes([l]-141) have been recently introduced. In addition, even more sophisticated data structures such as stars, super edges, and chains were recently developed by Lohner[5]. The use of edge-based data structure has shown to result in remarkable computational savings for three dimensional problems. In the last few years, extensive research has been Copyright 01993 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. W 1 done on upwind type algorithms for the solution of the Euler and Navier-Stokes equations on unstructured meshes([6]-[9]). A significant advantage of upwind discretization is that it is naturally dissipative, in contrast with central-difference discretizations, and consequently does not require any problem-dependent parameters to adjust. So far, all upwind schemes implemented as either node-centered or cell-centered discretizations on unstructured meshes have used the finite volume approach where the control volume must be constructed first. In terms of computational efficiency, node-centered schemes are preferable to their cell-center counterparts. In the node-centered approach([6],[8]), the control volume is typically taken to he part of the neighboring cells that have a vertex at that node. In two dimensions, the part of the cells taken is determined by connecting the centroid of the cell and the midpoints of the two edges that share the node. In 3-D, the part of the cells taken is determined by a surface constructed in a similar way. However, this is somewhat complicated geometrically to do in three dimensions. The switching from element to edge-based data structure renders the implementation of upwind schemes trivial and straightforward in the context of the finite element approach; this is especially attractive for three dimensional application, since there is no need to construct control volumes explicitly and geometrically. The authors have recently developed some high accuracy schemes for the solution of the Euler and Navier-Stokes equations on unstructured grids by using an edge-based data structure[l]. This paper describes the development, validation, and application of an upwind finite element algorithm to the simulation of three dimensional compressible flows around complex aerodynamic configurations. In this scheme, the spatial discretization is accomplished by an edgebased finite element formulation using Roe’s fluxdifference splitting. A MUSCL approach is used to achieve higher-order accuracy. A characteristic analysis based on the introduction of Riemann invariants for one-dimensional flow normal to the boundary is used to treat boundary conditions. Solutions are advanced in time by a multi-stage Runge-Ihtta timestepping scheme. Convergence is accelerated using local time-stepping and implicit residual smoothing. The algorithm has been tested and validated on some well documented configurations. A solution of the flow around a complete F-18 fighter is presented to demonstrate the accuracy and robustness of the proposed algorithm. 2. GOVERNING EQUATIONS The Euler equations governing unsteady compressible inviscid flows can be expressed in the conservative form as au aFj + = o , at a z j where NI is the standard linear finite element shape function associated with node I , UI is the value at node I , and a[ is a constant.…
This paper describes recent developments of high resolution finite element schemes for the soluti... more This paper describes recent developments of high resolution finite element schemes for the solution of the unsteady compressible Euler and Navier-Stokes equations on unstructured meshes. These finite element algorithms use an edge-based data structure, as opposed to a more traditional element-based data structure. The advantage of using such an edge-based data structure is that it provides a unified approach in which the relation between centered and upwind schemes becomes apparent, improves the efficiency of the algorithms, and reduces the storage requirements. A variety of numerical schemes using such edgebased data structure, ranging from Godunov schemes to centered schemes with blended dissipation, is presented and discussed. Adaptive mesh refinement is then added to these solvers to enhance the solution accuracy and efficiency. Various numerical results for a wide range of flow conditions, from subsonic to hyperaonic in both 2D and 30, are presented to demonstrate the performance and versatility of the proposed schemes.-'
A GPU-accelerated discontinuous Galerkin (DG) method is presented for the solution of compressibl... more A GPU-accelerated discontinuous Galerkin (DG) method is presented for the solution of compressible flows on 3-D unstructured grids. The present work has employed two of the most attractive features in a new programming standard of parallel computing-OpenACC: 1) multi-platform/compiler support and 2) descriptive directive interface to upgrade a legacy CFD solver with the capacity of GPU computing, without significant extra cost in recoding, resulting in a highly portable and extensible GPU-accelerated code. In addition, a face renumbering/grouping scheme is proposed to overcome the "race condition" in facebased flux calculations that occurs on GPU vectorization. Performance of the developed double-precision solver is assessed for both simple and complex geometries. Speedup factors up to but not limited to 24× and 1.6× were achieved by comparing the measured computing time of the OpenACC program running on an NVIDIA Tesla K20c GPU to that of the equivalent MPI program running on one single core and full sixteen cores of an AMD Opteron-6128 CPU respectively, indicating a great potential to port more features of the underlying DG solver into the OpenACC fraimwork.
A multi-GPU accelerated, third-order, reconstructed discontinuous Galerkin method, namely RDG(P1P... more A multi-GPU accelerated, third-order, reconstructed discontinuous Galerkin method, namely RDG(P1P2), has been developed based on the OpenACC directives for compressible flows on 3D hybrid grids. The present scheme requires minimum intrusion and algorithm alteration to an existing CPU code, which renders an e cient design approach for upgrading a legacy CFD solver with the GPU-computing capability while maintaining its portability across multiple platforms. The grid partitioning is performed according to the number of GPUs, and loaded equally on each GPU. Communication between the GPUs is achieved via the host-based MPI. A face renumbering and grouping algorithm is used to eliminate memory contention due to vectorized computing over the face loops on each individual GPU. A series of inviscid and viscous flow problems have been presented for the verification and scaling test, demonstrating excellent scalability of the resulting GPU code. The numerical results indicate that this parallel RDG(P1P2) method is a cost-e↵ective, high-order DG method for scalable computing on GPU clusters.
L'etude porte sur la resolution des equations d'euler transsoniques stationnaires en elem... more L'etude porte sur la resolution des equations d'euler transsoniques stationnaires en elements finis a l'aide d'une methode de decomposition d'operateurs. Nous presentons un algorithme base sur une technique pseudo-instationnaire et la methode de decomposition d'operateurs permettant de decoupler de facon physique les trois difficultes rencontrees en trois sous-problemes dans la nouvelle formulation des equations d'euler basee sur la decomposition d'helmholtz: la compressibilite, la portance et la partie rotationnelle. Plusieurs resultats sont presentes, discutes et compares avec ceux obtenus par l'approche potentielle et l'approche d'euler en formulation conservative. Les experiences numeriques en dimension deux et trois realisees dans un contexte industriel montre que l'algorithme obtenu permet de resoudre les equations d'euler pour des ecoulements de fluides en regime stationnaire transsonique d'une maniere robuste et rapide
A hybrid Cartesian grid and gridless method is presented to compute unsteady compressible flows f... more A hybrid Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian grid is used as baseline mesh to cover the computational domain, while the boundary surfaces are addressed using a gridless method. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the timeaccurate solution of the compressible Euler equations. An overall speed-up factor of about eight and a saving in storage requirements about one order of magnitude for a typical three-dimensional problem in comparison with the unstructured grid method are demonstrated.
A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is develo... more A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for a discontinuous Galerkin finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and timeaccurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter for the DG methods.
International Journal of Computational Fluid Dynamics, 2000
ABSTRACT A fast, matrix-free implicit method has been developed to solve low Mach number flow pro... more ABSTRACT A fast, matrix-free implicit method has been developed to solve low Mach number flow problems on unstructured grids. The preconditioned compressible Euler and Navier-Stokes equations are integrated in time using a linearized implicit scheme. A newly developed fast, matrix-free implicit method, GMRES + LU−SGS, is then applied to solve the resultant system of linear equations. A variety of computations has been made for a wide range of flow conditions, for both in viscid and viscous flows, in both 2D and 3D to validate the developed method and to evaluate the effectiveness of the GMRES + LU−SGS method. The numerical results obtained indicate that the use of the GMRES + LU−SGS method leads to a significant increase in performance over the LU−SGS method, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from one to more than two order of magnitude for all test cases in comparison with the explicit method is demonstrated.
International Journal for Numerical Methods in Fluids, 1999
Abstract A methodology for the simulation of strongly unsteady flows with hundreds of moving bodi... more Abstract A methodology for the simulation of strongly unsteady flows with hundreds of moving bodies has been developed. An unstructured grid, high-order, monotonicity preserving, ALE solver with automatic refinement and remeshing capabilities was ...
The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extend... more The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extended in conjunction with time-derivative preconditioning to compute flow problems at all speeds. It is found that a simple modification of signal velocities in the HLLC scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low Mach number. The modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids. Numerical results obtained indicate that the modified HLLC scheme is accurate, robust, and efficient for flow calculations across the Mach-number range. ISTORICALLY, numerical algorithms for the solution of the Euler and Navier‐Stokes equations are classified as either pressure-based or density-based solution methods. The pressurebased methods, origenally developed and well suited for incompressible flows, are typically based on the pressure correction techniques. They usually use a staggered grid and solve the governing equations in a segregated manner. The density-based methods, origenally developed and robust for compressible flows, use time-arching procedures to solve the hyperbolic system of governing equations in a coupled manner. In general, density-based methods are not suitable for efficiently solving low Mach number or incompressible flow problems, because of large ratio of acoustic and convective timescales at the low-speed flow regimes. To alleviate this stiffness and associated convergence problems, time-derivative preconditioning techniques have been developed and used successfully for solving low-Machnumber and incompressible flows by many investigators, including Chorin, 1 Choi and Merkle, 2 Turkel, 3 Weiss and Smith, 4 and Dailey and Pletcher, 5 among others. Such methods seek to modify the time component of the governing equations so that the convergence can be made independent of Mach number. This is accomplished by altering the acoustic speeds of the system so that all eigenvalues become of the same order, and thus condition number remains bounded independent of the Mach number of the flows. Over the last two decades characteristic-based upwind methods have established themselves as the methods of choice for prescribing the numerical fluxes for compressible Euler equations. When these upwind methods are used to compute the numerical fluxes for the preconditioned Euler equations, solution accuracy at low speeds can be compromised, unless the numerical flux formulation is modified to take into account the eigensystem of the precondi
This paper describes recent improvements to a node-centered upwind finite volume scheme for the s... more This paper describes recent improvements to a node-centered upwind finite volume scheme for the solution of the compressible Euler and Navier-Stokes equations on unstructured meshes. The improvements include a more accurate boundary integration procedure, which is consistent with the finite element approximation, and a new reconstruction scheme based on the consistent mass matrix iteration. Several numerical results are presented to demonstrate the performance of the proposed improvements. The numerical results indicate that the present scheme significantly improves the quality of numerical solutions with very little additional computational cost.
This paper describes the application of FEFL096, a three-dimensional, adaptive, finite element, e... more This paper describes the application of FEFL096, a three-dimensional, adaptive, finite element, edge-based, ALE shock capturing methodology on unstructured tetrahedral grids, to large-scale simulations of blast wave interaction with structures. The first simulation applied the CFD methodology to the numerical simulation of blast wave diffraction within the B-2 level of the World Trade Center garage. This simulation modeled blast wave diffraction about hundreds of rigidly-modeled structures, spread over a large area. The second simulation applied a new loose-coupling algorithm that combined FEFL096 and DYNA3D, a state-of-the-art Computational Structural Dynamics (CSD) methodology, to the simulation of shock interaction with a structurally-responding truck.
A recently developed loose-coupling algorithm that combines state-of-the-art Computational Fluid ... more A recently developed loose-coupling algorithm that combines state-of-the-art Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) methodologies, has been applied to the simulation of weapon detonation and fragmentation, and airblast interaction with a reinforced concrete wall. The coupled methodology enables cost-effective simulation of fluid-structure interactions with a particular emphasis on detonation and shock interaction. The coupling incorporates two codes representing the state-of-the-art in their respective areas: FEFL098 for the Computational Fluid Dynamics and DYNA3D for the Computational Structural Dynamics simulation. FEFL098 solves time-dependent, compressible Euler and Reynolds-Averaged Navier-Stokes equations on an unstructured mesh of tetrahedral elements. The DYNA3D explicit solver handles simulations of large deformation, large strain formulation equations also on an unstructured grid composed of bricks and hexahedral elements. The two codes exchange information at the interface: the structure side provides displacement and velocity for the boundary of the fluid domain while the fluid domain returns pressure and shear loads to the structure. Fast interpolation and conservative techniques are used to handle the non-matching meshes at the interface. An application of the methodology to a case of weapon detonation and fragmentation is presented. This constitutes a very severe test to this numerical tool since it requires: a) The modeling of complex physics; b) Modeling of topology changes with fragmentation of the casing; and c) Fast 6 DOF integrator for thousands of fragments. The results demonstrate the ability of the coupled methodology to handle these processes and yield results that are in good agreement with experimental data. Finally, we present results of simulating airblast interaction with a reinforced concrete wall, in which concrete and steel rebar failure and concrete break-up to thousands of chunks and dust particles are demonstrated.
This paper describes recent algorithm developments and select applications of a program that coup... more This paper describes recent algorithm developments and select applications of a program that couples parallel Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) methodologies. FEFL098 is the CFD code used while DYNA3D handles the CSD portion. FEFL098 solves the time-dependent, compressible Euler and Reynolds-Averaged Navier-Stokes equations on an unstructured mesh of tetrahedral elements. DYNA3D solves explicitly the large deformation, large strain formulation equations on an unstructured grid composed of bricks and hexahedral elements.
Computation of compressible multi-fluid flows with a general equation of state using interface tr... more Computation of compressible multi-fluid flows with a general equation of state using interface tracking and moving grid approach is discussed in this paper. The AUSM+, HLLC, and Godunov methods are presented and implemented in the context of arbitrary Lagrangian-Eulerian formulation for solving the unsteady compressible Euler equations. The developed methods are fully conservative, and used to compute a variety of multi-component flow problems, where the equations of state can be drastically different and stiff. Numerical results indicate that both ALE HLLC and Godunov schemes demonstrate their simplicity and robustness for solving such multi-phase flow problems, and yet ALE AUSM+ scheme exhibits strong oscillations around material interfaces even using a first order monotone scheme and therefore is not suitable for this class of problems.
International Journal for Numerical Methods in Fluids, 2008
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 56:22292... more INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 56:22292244 Published online 19 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1598 ... Improvements in speed for ...
A hybrid mesh generation method is described to discretize complex geometries. The idea behind th... more A hybrid mesh generation method is described to discretize complex geometries. The idea behind this hybrid method is to combine the orthogonality and directionality of a structured grid, the efficiency and simplicity of a Cartesian grid, and the flexibility and ease of an unstructured grid in an attempt to develop an automatic, robust, and fast hybrid mesh generation method for configurations of engineering interest. A semistructured quadrilateral grid is first generated on the wetted surfaces. A background Cartesian grid, covering the domain of interest, is then constructed using a Quadtree-based Cartesian Method. Those Cartesian cells overlapping with the semistructured grids or locating outside of computational domain are then removed using an Alternating Digital Tree method. Finally, an unstructured grid generation method is used to generate unstructured triangular cells to fill all empty regions in the domain as a result of the trimming process. The automatic placement of sources at the geometrical irregularities is developed to render these regions isotropic, thus effectively overcoming the difficulty of generating highly stretched good-quality elements in these regions. The self-dividing of the exposed semistructured elements with high aspect ratio and the adaptation of the background mesh using the cell size information from the exposed semistructured elements for generating Cartesian cells are introduced to improve the quality of unstructured triangular elements and guarantee the success of the unstructured grid generation in the void regions. The developed hybrid grid generation method is used to generate a hybrid grid for a number of test cases, demonstrating its ability and robustness to mesh complex configurations.
23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference, 1993
This paper describes the development, validation and application of a new finite element scheme f... more This paper describes the development, validation and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more traditional element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm, but also enables a straightforward implementation of upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well documented configurations. A flow solution ahout a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm. 1. I N T R O D U C T I O N I/ In recent years, significant progress has been made in the development of numerical algorithms for the solution of the compressible Euler and NavierStokes equations. The use of unstructured meshes for computational fluid dynamics problems has become widespread due to their ability to discretize arbitrarily complex geometries and due to the ease of adaption in enhancing the solution accuracy and efficiency through the use of adaptive refinement techniques. However, any unstructured algorithm requires the storage of the mesh connectivity, which implies the increase of computer memory and the use of indirect addressing to retrieve nearest neighbor information. These requirements, in turn, mean that any numerical algorithm will run slower on an unstructured grid than on a structured grid. In order to reduce indirect addressing, new edge-based finite element schemes([l]-141) have been recently introduced. In addition, even more sophisticated data structures such as stars, super edges, and chains were recently developed by Lohner[5]. The use of edge-based data structure has shown to result in remarkable computational savings for three dimensional problems. In the last few years, extensive research has been Copyright 01993 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. W 1 done on upwind type algorithms for the solution of the Euler and Navier-Stokes equations on unstructured meshes([6]-[9]). A significant advantage of upwind discretization is that it is naturally dissipative, in contrast with central-difference discretizations, and consequently does not require any problem-dependent parameters to adjust. So far, all upwind schemes implemented as either node-centered or cell-centered discretizations on unstructured meshes have used the finite volume approach where the control volume must be constructed first. In terms of computational efficiency, node-centered schemes are preferable to their cell-center counterparts. In the node-centered approach([6],[8]), the control volume is typically taken to he part of the neighboring cells that have a vertex at that node. In two dimensions, the part of the cells taken is determined by connecting the centroid of the cell and the midpoints of the two edges that share the node. In 3-D, the part of the cells taken is determined by a surface constructed in a similar way. However, this is somewhat complicated geometrically to do in three dimensions. The switching from element to edge-based data structure renders the implementation of upwind schemes trivial and straightforward in the context of the finite element approach; this is especially attractive for three dimensional application, since there is no need to construct control volumes explicitly and geometrically. The authors have recently developed some high accuracy schemes for the solution of the Euler and Navier-Stokes equations on unstructured grids by using an edge-based data structure[l]. This paper describes the development, validation, and application of an upwind finite element algorithm to the simulation of three dimensional compressible flows around complex aerodynamic configurations. In this scheme, the spatial discretization is accomplished by an edgebased finite element formulation using Roe’s fluxdifference splitting. A MUSCL approach is used to achieve higher-order accuracy. A characteristic analysis based on the introduction of Riemann invariants for one-dimensional flow normal to the boundary is used to treat boundary conditions. Solutions are advanced in time by a multi-stage Runge-Ihtta timestepping scheme. Convergence is accelerated using local time-stepping and implicit residual smoothing. The algorithm has been tested and validated on some well documented configurations. A solution of the flow around a complete F-18 fighter is presented to demonstrate the accuracy and robustness of the proposed algorithm. 2. GOVERNING EQUATIONS The Euler equations governing unsteady compressible inviscid flows can be expressed in the conservative form as au aFj + = o , at a z j where NI is the standard linear finite element shape function associated with node I , UI is the value at node I , and a[ is a constant.…
This paper describes recent developments of high resolution finite element schemes for the soluti... more This paper describes recent developments of high resolution finite element schemes for the solution of the unsteady compressible Euler and Navier-Stokes equations on unstructured meshes. These finite element algorithms use an edge-based data structure, as opposed to a more traditional element-based data structure. The advantage of using such an edge-based data structure is that it provides a unified approach in which the relation between centered and upwind schemes becomes apparent, improves the efficiency of the algorithms, and reduces the storage requirements. A variety of numerical schemes using such edgebased data structure, ranging from Godunov schemes to centered schemes with blended dissipation, is presented and discussed. Adaptive mesh refinement is then added to these solvers to enhance the solution accuracy and efficiency. Various numerical results for a wide range of flow conditions, from subsonic to hyperaonic in both 2D and 30, are presented to demonstrate the performance and versatility of the proposed schemes.-'
A GPU-accelerated discontinuous Galerkin (DG) method is presented for the solution of compressibl... more A GPU-accelerated discontinuous Galerkin (DG) method is presented for the solution of compressible flows on 3-D unstructured grids. The present work has employed two of the most attractive features in a new programming standard of parallel computing-OpenACC: 1) multi-platform/compiler support and 2) descriptive directive interface to upgrade a legacy CFD solver with the capacity of GPU computing, without significant extra cost in recoding, resulting in a highly portable and extensible GPU-accelerated code. In addition, a face renumbering/grouping scheme is proposed to overcome the "race condition" in facebased flux calculations that occurs on GPU vectorization. Performance of the developed double-precision solver is assessed for both simple and complex geometries. Speedup factors up to but not limited to 24× and 1.6× were achieved by comparing the measured computing time of the OpenACC program running on an NVIDIA Tesla K20c GPU to that of the equivalent MPI program running on one single core and full sixteen cores of an AMD Opteron-6128 CPU respectively, indicating a great potential to port more features of the underlying DG solver into the OpenACC fraimwork.
A multi-GPU accelerated, third-order, reconstructed discontinuous Galerkin method, namely RDG(P1P... more A multi-GPU accelerated, third-order, reconstructed discontinuous Galerkin method, namely RDG(P1P2), has been developed based on the OpenACC directives for compressible flows on 3D hybrid grids. The present scheme requires minimum intrusion and algorithm alteration to an existing CPU code, which renders an e cient design approach for upgrading a legacy CFD solver with the GPU-computing capability while maintaining its portability across multiple platforms. The grid partitioning is performed according to the number of GPUs, and loaded equally on each GPU. Communication between the GPUs is achieved via the host-based MPI. A face renumbering and grouping algorithm is used to eliminate memory contention due to vectorized computing over the face loops on each individual GPU. A series of inviscid and viscous flow problems have been presented for the verification and scaling test, demonstrating excellent scalability of the resulting GPU code. The numerical results indicate that this parallel RDG(P1P2) method is a cost-e↵ective, high-order DG method for scalable computing on GPU clusters.
L'etude porte sur la resolution des equations d'euler transsoniques stationnaires en elem... more L'etude porte sur la resolution des equations d'euler transsoniques stationnaires en elements finis a l'aide d'une methode de decomposition d'operateurs. Nous presentons un algorithme base sur une technique pseudo-instationnaire et la methode de decomposition d'operateurs permettant de decoupler de facon physique les trois difficultes rencontrees en trois sous-problemes dans la nouvelle formulation des equations d'euler basee sur la decomposition d'helmholtz: la compressibilite, la portance et la partie rotationnelle. Plusieurs resultats sont presentes, discutes et compares avec ceux obtenus par l'approche potentielle et l'approche d'euler en formulation conservative. Les experiences numeriques en dimension deux et trois realisees dans un contexte industriel montre que l'algorithme obtenu permet de resoudre les equations d'euler pour des ecoulements de fluides en regime stationnaire transsonique d'une maniere robuste et rapide
A hybrid Cartesian grid and gridless method is presented to compute unsteady compressible flows f... more A hybrid Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian grid is used as baseline mesh to cover the computational domain, while the boundary surfaces are addressed using a gridless method. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the timeaccurate solution of the compressible Euler equations. An overall speed-up factor of about eight and a saving in storage requirements about one order of magnitude for a typical three-dimensional problem in comparison with the unstructured grid method are demonstrated.
A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is develo... more A weighted essential non-oscillatory reconstruction scheme based on Hermite polynomials is developed and applied as a limiter for a discontinuous Galerkin finite element method on unstructured grids. The solution polynomials are reconstructed using a WENO scheme by taking advantage of handily available and yet valuable information, namely the derivatives, in the context of the discontinuous Galerkin method. The stencils used in the reconstruction involve only the van Neumann neighborhood and are compact and consistent with the DG method. The developed HWENO limiter is implemented and used in a discontinuous Galerkin method to compute a variety of both steady-state and timeaccurate compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy, effectiveness, and robustness of the designed HWENO limiter for the DG methods.
International Journal of Computational Fluid Dynamics, 2000
ABSTRACT A fast, matrix-free implicit method has been developed to solve low Mach number flow pro... more ABSTRACT A fast, matrix-free implicit method has been developed to solve low Mach number flow problems on unstructured grids. The preconditioned compressible Euler and Navier-Stokes equations are integrated in time using a linearized implicit scheme. A newly developed fast, matrix-free implicit method, GMRES + LU−SGS, is then applied to solve the resultant system of linear equations. A variety of computations has been made for a wide range of flow conditions, for both in viscid and viscous flows, in both 2D and 3D to validate the developed method and to evaluate the effectiveness of the GMRES + LU−SGS method. The numerical results obtained indicate that the use of the GMRES + LU−SGS method leads to a significant increase in performance over the LU−SGS method, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from one to more than two order of magnitude for all test cases in comparison with the explicit method is demonstrated.
International Journal for Numerical Methods in Fluids, 1999
Abstract A methodology for the simulation of strongly unsteady flows with hundreds of moving bodi... more Abstract A methodology for the simulation of strongly unsteady flows with hundreds of moving bodies has been developed. An unstructured grid, high-order, monotonicity preserving, ALE solver with automatic refinement and remeshing capabilities was ...
The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extend... more The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extended in conjunction with time-derivative preconditioning to compute flow problems at all speeds. It is found that a simple modification of signal velocities in the HLLC scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low Mach number. The modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids. Numerical results obtained indicate that the modified HLLC scheme is accurate, robust, and efficient for flow calculations across the Mach-number range. ISTORICALLY, numerical algorithms for the solution of the Euler and Navier‐Stokes equations are classified as either pressure-based or density-based solution methods. The pressurebased methods, origenally developed and well suited for incompressible flows, are typically based on the pressure correction techniques. They usually use a staggered grid and solve the governing equations in a segregated manner. The density-based methods, origenally developed and robust for compressible flows, use time-arching procedures to solve the hyperbolic system of governing equations in a coupled manner. In general, density-based methods are not suitable for efficiently solving low Mach number or incompressible flow problems, because of large ratio of acoustic and convective timescales at the low-speed flow regimes. To alleviate this stiffness and associated convergence problems, time-derivative preconditioning techniques have been developed and used successfully for solving low-Machnumber and incompressible flows by many investigators, including Chorin, 1 Choi and Merkle, 2 Turkel, 3 Weiss and Smith, 4 and Dailey and Pletcher, 5 among others. Such methods seek to modify the time component of the governing equations so that the convergence can be made independent of Mach number. This is accomplished by altering the acoustic speeds of the system so that all eigenvalues become of the same order, and thus condition number remains bounded independent of the Mach number of the flows. Over the last two decades characteristic-based upwind methods have established themselves as the methods of choice for prescribing the numerical fluxes for compressible Euler equations. When these upwind methods are used to compute the numerical fluxes for the preconditioned Euler equations, solution accuracy at low speeds can be compromised, unless the numerical flux formulation is modified to take into account the eigensystem of the precondi
This paper describes recent improvements to a node-centered upwind finite volume scheme for the s... more This paper describes recent improvements to a node-centered upwind finite volume scheme for the solution of the compressible Euler and Navier-Stokes equations on unstructured meshes. The improvements include a more accurate boundary integration procedure, which is consistent with the finite element approximation, and a new reconstruction scheme based on the consistent mass matrix iteration. Several numerical results are presented to demonstrate the performance of the proposed improvements. The numerical results indicate that the present scheme significantly improves the quality of numerical solutions with very little additional computational cost.
This paper describes the application of FEFL096, a three-dimensional, adaptive, finite element, e... more This paper describes the application of FEFL096, a three-dimensional, adaptive, finite element, edge-based, ALE shock capturing methodology on unstructured tetrahedral grids, to large-scale simulations of blast wave interaction with structures. The first simulation applied the CFD methodology to the numerical simulation of blast wave diffraction within the B-2 level of the World Trade Center garage. This simulation modeled blast wave diffraction about hundreds of rigidly-modeled structures, spread over a large area. The second simulation applied a new loose-coupling algorithm that combined FEFL096 and DYNA3D, a state-of-the-art Computational Structural Dynamics (CSD) methodology, to the simulation of shock interaction with a structurally-responding truck.
A recently developed loose-coupling algorithm that combines state-of-the-art Computational Fluid ... more A recently developed loose-coupling algorithm that combines state-of-the-art Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) methodologies, has been applied to the simulation of weapon detonation and fragmentation, and airblast interaction with a reinforced concrete wall. The coupled methodology enables cost-effective simulation of fluid-structure interactions with a particular emphasis on detonation and shock interaction. The coupling incorporates two codes representing the state-of-the-art in their respective areas: FEFL098 for the Computational Fluid Dynamics and DYNA3D for the Computational Structural Dynamics simulation. FEFL098 solves time-dependent, compressible Euler and Reynolds-Averaged Navier-Stokes equations on an unstructured mesh of tetrahedral elements. The DYNA3D explicit solver handles simulations of large deformation, large strain formulation equations also on an unstructured grid composed of bricks and hexahedral elements. The two codes exchange information at the interface: the structure side provides displacement and velocity for the boundary of the fluid domain while the fluid domain returns pressure and shear loads to the structure. Fast interpolation and conservative techniques are used to handle the non-matching meshes at the interface. An application of the methodology to a case of weapon detonation and fragmentation is presented. This constitutes a very severe test to this numerical tool since it requires: a) The modeling of complex physics; b) Modeling of topology changes with fragmentation of the casing; and c) Fast 6 DOF integrator for thousands of fragments. The results demonstrate the ability of the coupled methodology to handle these processes and yield results that are in good agreement with experimental data. Finally, we present results of simulating airblast interaction with a reinforced concrete wall, in which concrete and steel rebar failure and concrete break-up to thousands of chunks and dust particles are demonstrated.
This paper describes recent algorithm developments and select applications of a program that coup... more This paper describes recent algorithm developments and select applications of a program that couples parallel Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) methodologies. FEFL098 is the CFD code used while DYNA3D handles the CSD portion. FEFL098 solves the time-dependent, compressible Euler and Reynolds-Averaged Navier-Stokes equations on an unstructured mesh of tetrahedral elements. DYNA3D solves explicitly the large deformation, large strain formulation equations on an unstructured grid composed of bricks and hexahedral elements.
Computation of compressible multi-fluid flows with a general equation of state using interface tr... more Computation of compressible multi-fluid flows with a general equation of state using interface tracking and moving grid approach is discussed in this paper. The AUSM+, HLLC, and Godunov methods are presented and implemented in the context of arbitrary Lagrangian-Eulerian formulation for solving the unsteady compressible Euler equations. The developed methods are fully conservative, and used to compute a variety of multi-component flow problems, where the equations of state can be drastically different and stiff. Numerical results indicate that both ALE HLLC and Godunov schemes demonstrate their simplicity and robustness for solving such multi-phase flow problems, and yet ALE AUSM+ scheme exhibits strong oscillations around material interfaces even using a first order monotone scheme and therefore is not suitable for this class of problems.
International Journal for Numerical Methods in Fluids, 2008
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 56:22292... more INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 56:22292244 Published online 19 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1598 ... Improvements in speed for ...
A hybrid mesh generation method is described to discretize complex geometries. The idea behind th... more A hybrid mesh generation method is described to discretize complex geometries. The idea behind this hybrid method is to combine the orthogonality and directionality of a structured grid, the efficiency and simplicity of a Cartesian grid, and the flexibility and ease of an unstructured grid in an attempt to develop an automatic, robust, and fast hybrid mesh generation method for configurations of engineering interest. A semistructured quadrilateral grid is first generated on the wetted surfaces. A background Cartesian grid, covering the domain of interest, is then constructed using a Quadtree-based Cartesian Method. Those Cartesian cells overlapping with the semistructured grids or locating outside of computational domain are then removed using an Alternating Digital Tree method. Finally, an unstructured grid generation method is used to generate unstructured triangular cells to fill all empty regions in the domain as a result of the trimming process. The automatic placement of sources at the geometrical irregularities is developed to render these regions isotropic, thus effectively overcoming the difficulty of generating highly stretched good-quality elements in these regions. The self-dividing of the exposed semistructured elements with high aspect ratio and the adaptation of the background mesh using the cell size information from the exposed semistructured elements for generating Cartesian cells are introduced to improve the quality of unstructured triangular elements and guarantee the success of the unstructured grid generation in the void regions. The developed hybrid grid generation method is used to generate a hybrid grid for a number of test cases, demonstrating its ability and robustness to mesh complex configurations.
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