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1.
H. W. Zheng N. Qin F. C. G. A. Nicolleau & C. Shu 《Communications In Computational Physics》2011,9(1):68-88
An anisotropic solution adaptive method based on unstructured quadrilateral
meshes for inviscid compressible flows is proposed. The data structure, the directional
refinement and coarsening, including the method for initializing the refined
new cells, for the anisotropic adaptive method are described. It provides efficient high
resolution of flow features, which are aligned with the original quadrilateral mesh
structures. Five different cases are provided to show that it could be used to resolve
the anisotropic flow features and be applied to model the complex geometry as well as
to keep a relative high order of accuracy on an efficient anisotropic mesh. 相似文献
2.
Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations
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Haiyang Gao Z. J. Wang & H. T. Huynh 《Communications In Computational Physics》2013,13(4):1013-1044
A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR
approach employs the differential form of the equation and accounts for the jumps
in flux values at the cell boundaries by a correction procedure. In addition to being
simple and economical, it unifies several existing methods including discontinuous
Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG),
and I-continuous approaches. The first three of these approaches, originally derived
using the integral formulation, were recast here in the CPR framework, whereas the
I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a
triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes
equations are shown to compare the various schemes as well as to demonstrate the
capability of the CPR approach. 相似文献
3.
High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes
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Yizhou Lu Jun Zhu Shengzhu Cui Zhenming Wang Linlin Tian & Ning Zhao 《Communications In Computational Physics》2023,33(5):1217-1239
In this paper, a new multi-resolution weighted essentially non-oscillatory
(MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO
limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter
uses information of the LDG solution essentially only within the troubled cell itself, to
build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the
highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could
maintain the original order of accuracy in smooth regions and could simultaneously
suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition
that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the
freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter
is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction
methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classicalviscous examples are given to show the good performance of this third-order LDG
method with associated MR-WENO limiter. 相似文献
4.
A High-Order Cell-Centered Discontinuous Galerkin Multi-Material Arbitrary Lagrangian-Eulerian Method
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Fang Qing Xijun Yu Zupeng Jia Meilan Qiu & Xiaolong Zhao 《Communications In Computational Physics》2020,28(4):1464-1501
In this paper, a high-order cell-centered discontinuous Galerkin (DG) multi-material arbitrary Lagrangian-Eulerian (MMALE) method is developed for compressible fluid dynamics. The MMALE method utilizes moment-of-fluid (MOF) interface
reconstruction technology to simulate multi-materials of immiscible fluids. It is an
explicit time-marching Lagrangian plus remap type. In the Lagrangian phase, an updated high-order discontinuous Galerkin Lagrangian method is applied for the discretization of hydrodynamic equations, and Tipton's pressure relaxation closure model
is used in the mixed cells. A robust moment-of-fluid interface reconstruction algorithm
is used to provide the information of the material interfaces for remapping. In the rezoning phase, Knupp's algorithm is used for mesh smoothing. For the remapping
phase, a high-order accurate remapping method of the cell-intersection-based type is
proposed. It can be divided into four stages: polynomial reconstruction, polygon intersection, integration, and detection of problematic cells and limiting. Polygon intersection is based on the "clipping and projecting" algorithm, and detection of problematic
cells depends on a troubled cell marker, and a posteriori multi-dimensional optimal order detection (MOOD) limiting strategy is used for limiting. Numerical tests are given
to demonstrate the robustness and accuracy of our method. 相似文献
5.
High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes
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In this paper, the second-order and third-order Runge-Kutta discontinuous
Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory
(WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO
limiter is an extension of a finite volume multi-resolution WENO scheme developed
in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new
WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF
indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and
third-order RKDG methods with the associated multi-resolution WENO limiters are
developed as examples for general high-order RKDG methods, which could maintain
the original order of accuracy in smooth regions and keep essentially non-oscillatory
property near strong discontinuities by gradually degrading from the optimal order
to the first order. The linear weights inside the procedure of the new multi-resolution
WENO limiters can be set as any positive numbers on the condition that they sum
to one. A series of polynomials of different degrees within the troubled cell itself
are applied in a WENO fashion to modify the DG solutions in the troubled cell on
tetrahedral meshes. These new WENO limiters are very simple to construct, and can
be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such
spatial reconstruction methodology improves the robustness in the simulation on the
same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes)
and three-dimensional tests are performed to demonstrate the good performance of
the RKDG methods with new multi-resolution WENO limiters. 相似文献
6.
High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing
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Wei Wang Chi-Wang Shu H. C. Yee Dmitry V. Kotov & Bjö rn Sjö green 《Communications In Computational Physics》2015,17(2):317-336
In this paper, we extend the high order finite-difference method with subcell
resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and
multireaction inviscid chemical reactive flows, which are significantly more difficult
because of the multiple scales generated by different reactions. For reaction problems,
when the reaction time scale is very small, the reaction zone scale is also small and
the governing equations become very stiff. Wrong propagation speed of discontinuity
may occur due to the underresolved numerical solution in both space and time.
The present SR method for reactive Euler system is a fractional step method. In the
convection step, any high order shock-capturing method can be used. In the reaction
step, an ODE solver is applied but with certain computed flow variables in the shock
region modified by the Harten subcell resolution idea. Several numerical examples
of multispecies and multireaction reactive flows are performed in both one and two
dimensions. Studies demonstrate that the SR method can capture the correct propagation
speed of discontinuities in very coarse meshes. 相似文献
7.
A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh
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Marc R. J. Charest Clinton P. T. Groth & Pierre Q. Gauthier 《Communications In Computational Physics》2015,17(3):615-656
High-order discretization techniques offer the potential to significantly reduce
the computational costs necessary to obtain accurate predictions when compared
to lower-order methods. However, efficient and universally-applicable high-order
discretizations remain somewhat illusive, especially for more arbitrary unstructured
meshes and for incompressible/low-speed flows. A novel, high-order, central essentially
non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for
the solution of the conservation equations of viscous, incompressible flows on three-dimensional
unstructured meshes. Similar to finite element methods, coordinate transformations
are used to maintain the scheme's order of accuracy even when dealing
with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied
to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes
equations and the resulting discretized equations are solved with a parallel implicit
Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach
is adopted and the resulting temporal derivatives are discretized using the family of
high-order backward difference formulas (BDF). The proposed finite-volume scheme
for fully unstructured mesh is demonstrated to provide both fast and accurate solutions
for steady and unsteady viscous flows. 相似文献
8.
Marie Billaud Friess Jé rô me Breil Pierre-Henri Maire & Mikhail Shashkov 《Communications In Computational Physics》2014,15(2):330-364
In this paper we present recent developments concerning a Cell-Centered
Arbitrary Lagrangian Eulerian (CCALE) strategy using the Moment Of Fluid (MOF)
interface reconstruction for the numerical simulation of multi-material compressible
fluid flows on unstructured grids in cylindrical geometries. Especially, our attention
is focused here on the following points. First, we propose a new formulation of the
scheme used during the Lagrangian phase in the particular case of axisymmetric geometries. Then, the MOF method is considered for multi-interface reconstruction in
cylindrical geometry. Subsequently, a method devoted to the rezoning of polar meshes
is detailed. Finally, a generalization of the hybrid remapping to cylindrical geometries
is presented. These explorations are validated by mean of several test cases using unstructured grid that clearly illustrate the robustness and accuracy of the new method. 相似文献
9.
D. Wang 《Communications In Computational Physics》2006,1(4):716-735
A novel method for the generation of unstructured triangular surface meshes is presented. The method is based on remeshing techniques including edge splitting/contraction and edge swapping. Normalized edge lengths, based on a metric derived from curvature or from a user–specified spacing, are employed as the remeshing criterion. It is assumed that the geometry is input in the form of composite parametric surfaces, with Ferguson or Nurbs type multiple patch representation. Examples involving typical aircraft geometries and a ship model, are included to demonstrate how high quality meshes can be efficiently generated on surfaces with a high degree of geometric complexity. 相似文献
10.
A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations
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Based on the high order essentially non-oscillatory (ENO) Lagrangian type scheme on quadrilateral meshes presented in our earlier work [3], in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics. The main purpose of this work is to demonstrate our claim in [3] that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges, which restricts the accuracy of the resulting scheme to at most second order. The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes. Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties. 相似文献
11.
Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters: Three-Dimensional Unstructured Meshes
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This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions. 相似文献
12.
Adaptive Anisotropic Unstructured Mesh Generation Method Based on Fluid Relaxation Analogy
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In this paper, we extend the method (Fu et al., [1]) to anisotropic meshes by introducing an adaptive SPH (ASPH) concept with ellipsoidal kernels. First, anisotropic target feature-size and density functions, taking into account the effects of singularities, are defined based on the level-set methodology. Second, ASPH is developed such that the particle distribution relaxes towards the target functions. In order to prevent SPH particles from escaping the mesh generation regions, a ghost surface particle method is proposed in combination with a tailored interaction strategy. Necessary adaptations of supporting numerical algorithms, such as fast neighbor search, for enforcing mesh anisotropy are addressed. Finally, unstructured meshes are generated by an anisotropic Delaunay triangulation conforming to the Riemannian metrics for the resulting particle configuration. The performance of the proposed method is demonstrated by a set of benchmark cases. 相似文献
13.
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for least-squares
regularization, compressive sampling recovery, and interpolatory reconstruction
are becoming standard tools used in a variety of applications. Selection of a collocation
mesh is frequently a challenge, but methods that construct geometrically unstructured
collocation meshes have shown great potential due to attractive theoretical
properties and direct, simple generation and implementation. We investigate properties
of these meshes, presenting stability and accuracy results that can be used as
guides for generating stochastic collocation grids in multiple dimensions. 相似文献
14.
A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes
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Yunlong Yu Yanzhong Yao Guangwei Yuan & Xingding Chen 《Communications In Computational Physics》2016,20(5):1405-1423
In this paper, a conservative parallel iteration scheme is constructed to solve
nonlinear diffusion equations on unstructured polygonal meshes. The design is based
on two main ingredients: the first is that the parallelized domain decomposition is
embedded into the nonlinear iteration; the second is that prediction and correction
steps are applied at subdomain interfaces in the parallelized domain decomposition
method. A new prediction approach is proposed to obtain an efficient conservative
parallel finite volume scheme. The numerical experiments show that our parallel
scheme is second-order accurate, unconditionally stable, conservative and has linear
parallel speed-up. 相似文献
15.
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
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In this article we present a new class of high order accurate ArbitraryEulerian-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. 相似文献
16.
Implicit Quadrature-Free Direct Reconstruction Method for Efficient Scale-Resolving Simulations
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The present study develops implicit physical domain-based discontinuous Galerkin (DG) methods for efficient scale-resolving simulations on mixed-curved
meshes. Implicit methods are essential to handle stiff systems in many scale-resolving
simulations of interests in computational science and engineering. The physical
domain-based DG method can achieve high-order accuracy using the optimal bases set
and preserve the required accuracy on non-affine meshes. When using the quadrature-based DG method, these advantages are overshadowed by severe computational costs
on mixed-curved meshes, making implicit scale-resolving simulations unaffordable.
To address this issue, the quadrature-free direct reconstruction method (DRM) is extended to the implicit DG method. In this approach, the generalized reconstruction
approximates non-linear flux functions directly in the physical domain, making the
computing-intensive numerical integrations precomputable at a preprocessing step.
The DRM operator is applied to the residual computation in the matrix-free method.
The DRM operator can be further extended to the system matrix computation for the
matrix-explicit Krylov subspace method and preconditioning. Finally, the A-stable
Rosenbrock-type Runge–Kutta methods are adopted to achieve high-order accuracy
in time. Extensive verification and validation from the manufactured solution to implicit large eddy simulations are conducted. The computed results confirm that the
proposed method significantly improves computational efficiency compared to the
quadrature-based method while accurately resolving detailed unsteady flow features
that are hardly captured by scale-modeled simulations. 相似文献
17.
Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme
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Pierre-Henri Maire Raphaë l Loubè re & Pavel Vá chal 《Communications In Computational Physics》2011,10(4):940-978
The aim of the present work is to develop a general formalism to derive
staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured
grids. To this end, we make use of the compatible discretization that has been initially
introduced by E. J. Caramana et al., in J. Comput. Phys., 146 (1998). Namely, momentum
equation is discretized by means of subcell forces and specific internal energy
equation is obtained using total energy conservation. The main contribution of this
work lies in the fact that the subcell force is derived invoking Galilean invariance and
thermodynamic consistency. That is, we deduce a general form of the sub-cell force so
that a cell entropy inequality is satisfied. The subcell force writes as a pressure contribution
plus a tensorial viscous contribution which is proportional to the difference
between the nodal velocity and the cell-centered velocity. This cell-centered velocity is
a supplementary degree of freedom that is solved by means of a cell-centered approximate
Riemann solver. To satisfy the second law of thermodynamics, the local subcell
tensor involved in the viscous part of the subcell force must be symmetric positive
definite. This subcell tensor is the cornerstone of the scheme. One particular expression
of this tensor is given. A high-order extension of this discretization is provided.
Numerical tests are presented in order to assess the efficiency of this approach. The
results obtained for various representative configurations of one- and two-dimensional
compressible fluid flows show the robustness and the accuracy of this scheme. 相似文献
18.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws
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Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
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 solving 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 not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but 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 isless 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. 相似文献
19.
Gang Wang Xiangyang Cui & Guangyao Li 《Communications In Computational Physics》2016,20(5):1258-1282
It is well-known that the traditional full integral quadrilateral element fails
to provide accurate results to the Helmholtz equation with large wave numbers due to
the "pollution error" caused by the numerical dispersion. To overcome this deficiency,
this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic
problems by using quadrilateral element. In the present EDM, the quadrilateral
element is first subdivided into four sub-triangles, and the local acoustic gradient in
each sub-triangle is obtained using linear interpolation function. The acoustic gradient
field of the whole quadrilateral is then formulated through a weighted averaging
operation, which means only one integration point is adopted to construct the system
matrix. To cure the numerical instability of one-point integration, a variation gradient
item is complemented by variance of the local gradients. The discretized system
equations are derived using the generalized Galerkin weak form. Numerical examples
demonstrate that the EDM can achieves better accuracy and higher computational efficiency.
Besides, as no mapping or coordinate transformation is involved, restrictions
on the shape elements can be easily removed, which makes the EDM works well even
for severely distorted meshes. 相似文献
20.
Håkan Hallberg Richard Lewin Anna Elander Emma Hansson 《Journal of plastic surgery and hand surgery》2018,52(4):253-258
In recent years, it has become increasingly popular to use matrices, such as acellular dermal matrices, in implant-based breast reconstruction. To lower the cost and to avoid implanting biological material, the use of synthetic meshes has been proposed. This is the first study examining TIGR® Mesh in a larger series of immediate breast reconstruction. The aims of the study were to examine complications and predictors for complications. All consecutive patients operated on with breast reconstruction with TIGR® Matrix Surgical Mesh and tissue expanders (TEs) or permanent implant between March 2015 and September 2016 in our department were prospectively included. Exclusion criteria were ongoing smoking, BMI (kg/m2)?>?30, planned postoperative radiation, and inability to leave informed consent. Fifteen breasts (23%) were affected by complications within 30 d: four (6.2%) major complications and eleven (17%) minor complications. The major complications included two implant losses and one pulmonary embolism (PE). Predictors for a complication were age over 51 years, BMI over 24.5?kg/m2, large resection weight, and the need for a wise pattern excision of skin. Four minor surgical complications occurred after 30 d (minimum follow-up 17 months). There were no implant losses. In addition, minor aesthetic corrections, such as dog-ear resection, were performed in 10 breasts. In conclusion, breast reconstruction with a TE in combination with TIGR® Matrix Surgical Mesh can be performed with a low complication rate. 相似文献