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1.
A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh 下载免费PDF全文
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
2.
The formation of singularities in relativistic flows is not well understood.
Smooth solutions to the relativistic Euler equations are known to have a finite lifespan;
the possible breakdown mechanisms are shock formation, violation of the subluminal
conditions and mass concentration. We propose a new hybrid Glimm/central-upwind
scheme for relativistic flows. The scheme is used to numerically investigate,
for a family of problems, which of the above mechanisms is involved. 相似文献
3.
Liang Wang Jianchun Mi Xuhui Meng & Zhaoli Guo 《Communications In Computational Physics》2015,17(4):908-924
A mass-conserving lattice Boltzmann model based on the Bhatnagar-Gross-Krook
(BGK) model is proposed for non-Newtonian fluid flows. The equilibrium distribution
function includes the local shear rate related with the viscosity and a variable
parameter changing with the shear rate. With the additional parameter, the relaxation
time in the collision can be fixed invariable to the viscosity. Through the Chapman-Enskog
analysis, the macroscopic equations can be recovered from the present mass-conserving
model. Two flow problems are simulated to validate the present model
with a local computing scheme for the shear rate, and good agreement with analytical
solutions and/or other published results are obtained. The results also indicate that
the present modified model is more applicable to practical non-Newtonian fluid flows
owing to its better accuracy and more robustness than previous methods. 相似文献
4.
In this paper, we present an adaptive moving mesh technique for solving
the incompressible viscous flows using the vorticity stream-function formulation. The
moving mesh strategy is based on the approach proposed by Li et al. [J. Comput. Phys.,
170 (2001), pp. 562–588] to separate the mesh-moving and evolving PDE at each time
step. The Navier-Stokes equations are solved in the vorticity stream-function form by
a finite-volume method in space, and the mesh-moving part is realized by solving the
Euler-Lagrange equations to minimize a certain variation in conjunction with a more
sophisticated monitor function. A conservative interpolation is used to redistribute
the numerical solutions on the new meshes. This paper discusses the implementation
of the periodic boundary conditions, where the physical domain is allowed to deform
with time while the computational domain remains fixed and regular throughout. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm. 相似文献
5.
This paper summarizes suitable material models for creep and damage of concrete which are coupled with heat and moisture transfer. The fully coupled approach or the staggered coupling is assumed. Governing equations are spatially discretized by the finite element method and the temporal discretization is done by the generalized trapezoidal method. Systems of non-linear algebraic equations are solved by the Newton method. Development of an efficient and extensible computer code based on the C++ programming language is described. Finally, successful analyses of two real engineering problems are described. 相似文献
6.
Liang Pan Guiping Zhao Baolin Tian & Shuanghu Wang 《Communications In Computational Physics》2013,14(5):1347-1371
In this paper, a gas kinetic scheme for the compressible multicomponent
flows is presented by making use of two-species BGK model in [A. D. Kotelnikov and
D. C. Montgomery, A Kinetic Method for Computing Inhomogeneous Fluid Behavior,
J. Comput. Phys. 134 (1997) 364-388]. Different from the conventional BGK model,
the collisions between different species are taken into consideration. Based on the
Chapman-Enskog expansion, the corresponding macroscopic equations are derived
from this two-species model. Because of the relaxation terms in the governing equations, the method of operator splitting is applied. In the hyperbolic part, the integral
solutions of the BGK equations are used to construct the numerical fluxes at the cell
interface in the framework of finite volume method. Numerical tests are presented
in this paper to validate the current approach for the compressible multicomponent
flows. The theoretical analysis on the spurious oscillations at the interface is also presented. 相似文献
7.
A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime 下载免费PDF全文
We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated
from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to
the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically
symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based on the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level.
Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear
interacting wave patterns. As an application, we investigate the late-time asymptotics
of perturbed steady solutions and demonstrate its convergence for late time toward
another steady solution, taking the overall effect of the perturbation into account. 相似文献
8.
S. C. Fu R. M. C. So & W. W. F. Leung 《Communications In Computational Physics》2011,9(5):1257-1283
The objective of this paper is to seek an alternative to the numerical simulation
of the Navier-Stokes equations by a method similar to solving the BGK-type
modeled lattice Boltzmann equation. The proposed method is valid for both gas and
liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations
for two distribution functions; one for mass and another for thermal energy. These
equations are derived by considering an infinitesimally small control volume with a
velocity lattice representation for the distribution functions. The zero-order moment
equation of the mass distribution function is used to recover the continuity equation,
while the first-order moment equation recovers the linear momentum equation. The
recovered equations are correct to the first order of the Knudsen number (Kn); thus,
satisfying the continuum assumption. Similarly, the zero-order moment equation of
the thermal energy distribution function is used to recover the thermal energy equation.
For aerodynamic flows, it is shown that the finite difference solution of the DFS
is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model
and a specified equation of state. Thus formulated, the DFS can be used to simulate a
variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics,
compressible flow with shocks, incompressible isothermal and non-isothermal Couette
flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used
to demonstrate the validity and extent of the DFS. Very good to excellent agreement
with known analytical and/or numerical solutions is obtained; thus lending evidence
to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid
flow simulations. 相似文献
9.
Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning 下载免费PDF全文
Yufei Wang Ziju Shen Zichao Long & Bin Dong 《Communications In Computational Physics》2020,28(5):2158-2179
Conservation laws are considered to be fundamental laws of nature. It has
broad applications in many fields, including physics, chemistry, biology, geology, and
engineering. Solving the differential equations associated with conservation laws is a
major branch in computational mathematics. The recent success of machine learning,
especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as an
MDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on
1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner.
We will show that the problem of solving conservation laws can be naturally viewed
as a sequential decision-making process, and the numerical schemes learned in such a
way can easily enforce long-term accuracy. Furthermore, the learned policy network
is carefully designed to determine a good local discrete approximation based on the
current state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how to
discretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with some
state-of-the-art numerical solvers such as WENO schemes, and supervised learning
based approach L3D and PINN, and how well it generalizes. 相似文献
10.
A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations 下载免费PDF全文
Kaipeng Wang rew Christlieb Yan Jiang & Mengping Zhang 《Communications In Computational Physics》2021,29(1):237-264
In this paper, a class of high order numerical schemes is proposed to solve
the nonlinear parabolic equations with variable coefficients. This method is based on
our previous work [11] for convection-diffusion equations, which relies on a special
kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems.
To overcome these difficulties, a new kernel-based formulation is designed to approach
the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems,
hence allowing much larger time step evolution compared with other explicit schemes.
In addition, without extra computational cost, the proposed scheme can enlarge the
available interval of the special parameter in the formulation, leading to less errors
and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for
one- and two-dimensional scalar and system, demonstrating the designed high order
accuracy and unconditionally stable property of the scheme. 相似文献
11.
VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity 下载免费PDF全文
Yujie Liu & Chao Yang 《Communications In Computational Physics》2022,31(3):739-770
We present VPVnet, a deep neural network method for the Stokes’ equations under reduced regularity. Different with recently proposed deep learning methods [40,51] which are based on the original form of PDEs, VPVnet uses the least square
functional of the first-order velocity-pressure-vorticity (VPV) formulation ([30]) as loss
functions. As such, only first-order derivative is required in the loss functions, hence
the method is applicable to a much larger class of problems, e.g. problems with nonsmooth solutions. Despite that several methods have been proposed recently to reduce
the regularity requirement by transforming the original problem into a corresponding
variational form, while for the Stokes’ equations, the choice of approximating spaces
for the velocity and the pressure has to satisfy the LBB condition additionally. Here
by making use of the VPV formulation, lower regularity requirement is achieved with
no need for considering the LBB condition. Convergence and error estimates have
been established for the proposed method. It is worth emphasizing that the VPVnet
method is divergence-free and pressure-robust, while classical inf-sup stable mixed
finite elements for the Stokes’ equations are not pressure-robust. Various numerical
experiments including 2D and 3D lid-driven cavity test cases are conducted to demonstrate its efficiency and accuracy. 相似文献
12.
In this paper, we consider the problem of existence of certain global solutions for general discrete‐time backward nonlinear equations, defined on infinite dimensional ordered Banach spaces. This class of nonlinear equations includes as special cases many of the discrete‐time Riccati equations arising both in deterministic and stochastic optimal control problems. On the basis of a linear matrix inequalities approach, we give necessary and sufficient conditions for the existence of maximal, stabilizing, and minimal solutions of the considered discrete‐time backward nonlinear equations. As an application, we discuss the existence of stabilizing solutions for discrete‐time Riccati equations of stochastic control and filtering on Hilbert spaces. The tools provided by this paper show that a wide class of nonlinear equations can be treated in a uniform manner. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
13.
Chaotic dynamics and convergence analysis of temporal difference algorithms with bang‐bang control 下载免费PDF全文
Martin Brown 《Optimal control applications & methods.》2016,37(1):108-126
Reinforcement learning is a powerful tool used to obtain optimal control solutions for complex and difficult sequential decision making problems where only a minimal amount of a priori knowledge exists about the system dynamics. As such, it has also been used as a model of cognitive learning in humans and applied to systems, such as humanoid robots, to study embodied cognition. In this paper, a different approach is taken where a simple test problem is used to investigate issues associated with the value function's representation and parametric convergence. In particular, the terminal convergence problem is analyzed with a known optimal control policy where the aim is to accurately learn the value function. For certain initial conditions, the value function is explicitly calculated and it is shown to have a polynomial form. It is parameterized by terms that are functions of the unknown plant's parameters and the value function's discount factor, and their convergence properties are analyzed. It is shown that the temporal difference error introduces a null space associated with the finite horizon basis function during the experiment. The learning problem is only non‐singular when the experiment termination is handled correctly and a number of (equivalent) solutions are described. Finally, it is demonstrated that, in general, the test problem's dynamics are chaotic for random initial states and this causes digital offset in the value function learning. The offset is calculated, and a dead zone is defined to switch off learning in the chaotic region. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
14.
Shamsulhaq Basir 《Communications In Computational Physics》2023,33(5):1240-1269
This paper explores the difficulties in solving partial differential equations
(PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach is the
requirement for manual hyperparameter tuning, making it impractical in the absence
of validation data or prior knowledge of the solution. Our investigations of the loss
landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate. Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and
hinder convergence. To address these challenges, we introduce a novel method that
bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients. Consequently, we reduce the dimension of the
search space and make learning PDEs with non-smooth solutions feasible. Our method
also provides a mechanism to focus on complex regions of the domain. Besides, we
present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model’s prediction, with adaptive and independent
learning rates inspired by adaptive subgradient methods. We apply our approach to
solve various linear and non-linear PDEs. 相似文献
15.
A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids 下载免费PDF全文
Bassou Khouya Mofdi El-Amrani & Mohammed Seaid 《Communications In Computational Physics》2022,31(1):224-256
We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressible
Navier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite element
method for the space discretization. This class of computational solvers benefits from
the geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows using
time steps larger than its Eulerian counterparts. In the current study, we implement
three-dimensional limiters to convert the proposed solver to a fully mass-conservative
and essentially monotonicity-preserving method in addition of a low computational
cost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. The
proposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical results
illustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominated
flow problems on unstructured tetrahedral meshes. 相似文献
16.
A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations 下载免费PDF全文
Xiaodong Liu Yidong Xia Hong Luo & Lijun Xuan 《Communications In Computational Physics》2016,20(4):1016-1044
A comparative study of two classes of third-order implicit time integration
schemes is presented for a third-order hierarchical WENO reconstructed discontinuous
Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes
equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3)
scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential
algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme,
a remarkable feature of the ROW schemes is that, they only require one approximate
Jacobian matrix calculation every time step, thus considerably reducing the overall
computational cost. A variety of test cases, ranging from inviscid flows to DNS of
turbulent flows, are presented to assess the performance of these schemes. Numerical
experiments demonstrate that the third-order ROW scheme for the DAEs of index-2
can not only achieve the designed formal order of temporal convergence accuracy in
a benchmark test, but also require significantly less computing time than its ESDIRK3
counterpart to converge to the same level of discretization errors in all of the flow
simulations in this study, indicating that the ROW methods provide an attractive alternative
for the higher-order time-accurate integration of the unsteady compressible
Navier-Stokes equations. 相似文献
17.
John C. Morrison Scott Boyd Luis Marsano Bernard Bialecki Thomas Ericsson & Jose Paulo Santos 《Communications In Computational Physics》2009,5(5):959-985
The theory of domain decomposition is described and used to divide the variable domain of a diatomic molecule into separate regions which are solved independently. This approach makes it possible to use fast Krylov methods in the broad interior of the region while using explicit methods such as Gaussian elimination on the boundaries. As is demonstrated by solving a number of model problems, these methods enable one to obtain solutions of the relevant partial differential equations and eigenvalue equations accurate to six significant figures with a small amount of computational time. Since the numerical approach described in this article decomposes the variable space into separate regions where the equations are solved independently, our approach is very well-suited to parallel computing and offers the long term possibility of studying complex molecules by dividing them into smaller fragments that are calculated separately. 相似文献
18.
J. Vides B. Braconnier E. Audit C. Berthon & B. Nkonga 《Communications In Computational Physics》2014,15(1):46-75
We present a new numerical method to approximate the solutions of an
Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an
important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity
source term involved in the latter equations. In order to approximate this source term,
its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The
method has been implemented in the software HERACLES [29] and several numerical
experiments involving gravitational flows for astrophysics highlight the scheme. 相似文献
19.
A Hybrid Immersed Boundary-Lattice Boltzmann Method for Simulation of Viscoelastic Fluid Flows Interaction with Complex Boundaries 下载免费PDF全文
M. H. Sedaghat A. A. H. Bagheri M. M. Shahmardan M. Norouzi B. C. Khoo & P. G. Jayathilake 《Communications In Computational Physics》2021,29(5):1411-1445
In this study, a numerical technique based on the Lattice Boltzmann method
is presented to model viscoelastic fluid interaction with complex boundaries which are
commonly seen in biological systems and industrial practices. In order to accomplish
numerical simulation of viscoelastic fluid flows, the Newtonian part of the momentum
equations is solved by the Lattice Boltzmann Method (LBM) and the divergence of the
elastic tensor, which is solved by the finite difference method, is added as a force term
to the governing equations. The fluid-structure interaction forces are implemented
through the Immersed Boundary Method (IBM). The numerical approach is validated
for Newtonian and viscoelastic fluid flows in a straight channel, a four-roll mill geometry as well as flow over a stationary and rotating circular cylinder. Then, a numerical
simulation of Oldroyd-B fluid flow around a confined elliptical cylinder with different
aspect ratios is carried out for the first time. Finally, the present numerical approach
is used to simulate a biological problem which is the mucociliary transport process of
human respiratory system. The present numerical results are compared with appropriate analytical, numerical and experimental results obtained from the literature. 相似文献
20.
Upwind Biased Local RBF Scheme with PDE Centres for the Steady Convection Diffusion Equations with Continuous and Discontinuous Boundary Conditions 下载免费PDF全文
RBF based grid-free scheme with PDE centres is experimented in this work
for solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemes
can give converged solutions for the entire range of diffusivity wherein sharp layers are
expected in the solutions and accurate computation of these layers is a big challenge
for most of the numerical schemes. Radial Basis Function (RBF) based Local Hermitian
Interpolation (LHI) with PDE centres is one such integration scheme which has some
built in upwind effect and hence may be a good solver for the convection dominated
problems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based scheme
with PDE centres, which was initially used to solve some time dependent problems
in [10]. RBF based numerical schemes are one type of grid free numerical schemes
based on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a variety
of steady benchmark problems with continuous and discontinuous boundary data and
validated against the corresponding exact solutions. Comparisons of the solutions of
the convective dominant benchmark problems show that the upwind biasing either
in source centres or PDE centres gives convergent solutions consistently and is stable
without any oscillations especially for problems with discontinuities in the boundary
conditions. It is observed that the accuracy of the solutions is better than the solutions
of other standard integration schemes particularly when the computations are carried
out with fewer centers. 相似文献