共查询到20条相似文献,搜索用时 15 毫秒
1.
A Novel Iterative Penalty Method to Enforce Boundary Conditions in Finite Volume POD-Galerkin Reduced Order Models for Fluid Dynamics Problems 下载免费PDF全文
S. Kelbij Star Giovanni Stabile Francesco Belloni Gianluigi Rozza & Joris Degroote 《Communications In Computational Physics》2021,30(1):34-66
A Finite-Volume based POD-Galerkin reduced order model is developed for
fluid dynamics problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the lifting function method,
whose aim is to obtain homogeneous basis functions for the reduced basis space and
the penalty method where the boundary conditions are enforced in the reduced order
model using a penalty factor. The penalty method is improved by using an iterative
solver for the determination of the penalty factor rather than tuning the factor with a
sensitivity analysis or numerical experimentation.The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet
channels and one outlet channel. The results show that the boundaries of the reduced
order model can be controlled with the boundary control methods and the same order
of accuracy is achieved for the velocity and pressure fields. Finally, the reduced order
models are 270-308 times faster than the full order models for the lid driven cavity test
case and 13-24 times for the Y-junction test case. 相似文献
2.
Yulei Liao & Pingbing Ming 《Communications In Computational Physics》2021,29(5):1365-1384
We propose a new method to deal with the essential boundary conditions
encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory,
which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsche's variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs.
We prove the error estimate in the energy norm and illustrate the method on several
representative problems posed in at most 100 dimension. 相似文献
3.
In this paper we propose some Newton-type algorithms for the numerical solution of both unconstrained and constrained discrete-time optimal control problems. The approach followed here is based on a suitable augmented Lagrangian function whose unconstrained minimization yields the solution of the optimal control problem and the associated multipliers. We show that the Hessian matrix of the augmented Lagrangian function has a sparse structure which allows the use of an efficient decomposition scheme for the computation of the Newton's direction. In addition, consistent approximations of the Newton's direction are described. These approximations may allow a further reduction of the computational cost. Two numerical examples are reported. 相似文献
4.
A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems 下载免费PDF全文
In this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems. 相似文献
5.
The interaction of the two drugs warfarin and phenylbutazone has previously been considered as a time-optimal control problem with state inequality constraints. We include bounds for the control and show that necessary optimality conditions and junction conditions for bounded state variables lead to boundary value problems with switching and junction conditions. A special version of the multiple-shooting algorithm is employed for solving the different types of boundary value problems. The switching structure of the optimal control is determined for a range of parameters in the state constraint. Owing to the special structure of the control, a state space solution is obtained in a first step which reduces the numerical complexity of the problem. It is shown how the numerical results can be used to compute the generalized gradient of the optimal value function explicitly. 相似文献
6.
An integral approach to solve finite‐horizon optimal control problems posed by set‐point changes in electrochemical hydrogen reactions is developed. The methodology extends to nonlinear problems with regular, convex Hamiltonians that cannot be explicitly minimized, i.e. where the functional dependence of the H‐minimal control on the state and costate variables is not known. The Lagrangian functions determining trajectory costs will not have special restrictions other than positiveness, but for simplicity the final penalty will be assumed quadratic. The answer to the problem is constructed through the solution to a coupled system of three first‐order quasi‐linear partial differential equations (PDEs) for the missing boundary conditions x(T), λ(0) of the Hamiltonian equations, and for the final value of the control variable u(T). The independent variables of these PDEs are the time‐duration T of the process and the characteristic parameter S of the final penalty. The solution provides information on the whole (T, S)‐family of control problems, which can be used not only to construct the individual optimal control strategies online, but also for global design purposes. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
7.
F. Criado‐Aldeanueva F. Criado N. Odishelidze J. M. Sanchez 《Optimal control applications & methods.》2010,31(6):497-503
In this paper, the optimal control problem for the Helmholtz equation with non‐local boundary conditions is considered. The necessary and sufficient conditions of optimality in a maximum principle form have been obtained. We note that this problem is basically different from classical‐type problems because it is impossible to use Green's formula and we cannot rewrite it in the variational form widely used in the literature. So it is impossible to use all the theory that has been developed for optimal control problems with classical boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
8.
A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions 下载免费PDF全文
In this paper, we study a lattice Boltzmann method for the advection-diffusion equation with Neumann boundary conditions on general boundaries. A
novel mass conservative scheme is introduced for implementing such boundary conditions, and is analyzed both theoretically and numerically.Second order convergence is predicted by the theoretical analysis, and numerical
investigations show that the convergence is at or close to the predicted rate. The numerical investigations include time-dependent problems and a steady-state diffusion
problem for computation of effective diffusion coefficients. 相似文献
9.
In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact. 相似文献
10.
Computational Investigation of the Effects of Sample Geometry on the Superconducting-Normal Phase Boundary and Vortex-Antivortex States in Mesoscopic Superconductors 下载免费PDF全文
Sangbum Kim Max Gunzburger Janet Peterson & Chia-Ren Hu 《Communications In Computational Physics》2009,6(4):673-698
A computational study of superconducting states near the superconducting-normal
phase boundary in mesoscopic finite cylinders is presented. The computational
approach uses a finite element method to find numerical solutions of the linearized
Ginzburg-Landau equation for samples with various sizes, aspect ratios, and cross-sectional
shapes, i.e., squares, triangles, circles, pentagons, and four star shapes. The
vector potential is determined using a finite element method with two penalty terms
to enforce the gauge conditions that the vector potential is solenoidal and its normal
component vanishes at the surface(s) of the sample. The eigenvalue problem for the
linearized Ginzburg-Landau equations with homogeneous Neumann boundary conditions
is solved and used to construct the superconducting-normal phase boundary
for each sample. Vortex-antivortex (V-AV) configurations for each sample that accurately
reflect the discrete symmetry of each sample boundary were found through the
computational approach. These V-AV configurations are realized just within the phase
boundary in the magnetic field-temperature phase diagram. Comparisons are made
between the results obtained for the different sample shapes. 相似文献
11.
Achim Ilchmann Leslie Leben Jonas Witschel Karl Worthmann 《Optimal control applications & methods.》2019,40(2):351-366
We study the optimal control problem (OCP) for regular linear differentialalgebraic systems. To this end, we introduce the input index, which allows, on the one hand, to characterize the space of consistent initial values in terms of a Kalman‐like matrix and, on the other hand, the necessary smoothness properties of the control. The latter is essential to make the problem accessible from a numerical point of view. Moreover, we derive an augmented system as the key to analyze the OCP with tools well known from optimal control of ordinary differential equations. The new concepts of the input index and the augmented system provide easily checkable sufficient conditions, which ensure that the stage costs are consistent with the differential‐algebraic system. 相似文献
12.
J. Haslinger P. Neittaanmki T. Tiihonen A. Kaarna 《Optimal control applications & methods.》1988,9(2):127-144
In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included. 相似文献
13.
A. Q. Xing Z. H. Chen C. L. Wang Y. Y. Yao 《Optimal control applications & methods.》1989,10(2):173-180
Given the well known concept that optimal control problems may be solved either by the maximum principle or by the dynamic programming technique which employs many numerical algorithms, this paper attempts to show that the exact penalty function method may be used to transform a constrained optimal control problem into an unconstrained optimal control problem. Under certain conditions the constrained optimal control problem is shown to be equivalent to an unconstrained optimal control problem, which can be easily solved by a numerical technique. 相似文献
14.
An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications 下载免费PDF全文
Wei-Fan Hu Te-Sheng Lin Yu-Hau Tseng & Ming-Chih Lai 《Communications In Computational Physics》2023,33(4):1090-1105
A new and efficient neural-network and finite-difference hybrid method is
developed for solving Poisson equation in a regular domain with jump discontinuities
on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional
treatment accounting for the jump discontinuities must be employed. Here, we aim to
elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The
neural network learning machinery incorporating the given jump conditions finds the
singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy
to implement and efficient. The two- and three-dimensional numerical results show
that the present hybrid method preserves second-order accuracy for the solution and
its derivatives, and it is comparable with the traditional immersed interface method in
the literature. As an application, we solve the Stokes equations with singular forces to
demonstrate the robustness of the present method. 相似文献
15.
Splitting Finite Difference Methods on Staggered Grids for the Three-Dimensional Time-Dependent Maxwell Equations 下载免费PDF全文
In this paper, we study splitting numerical methods for the three-dimensional
Maxwell equations in the time domain. We propose a new kind of splitting finite-difference time-domain schemes on a staggered grid, which consists of only two stages
for each time step. It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary
conditions. Both numerical dispersion analysis and numerical experiments are also
presented to illustrate the efficiency of the proposed schemes. 相似文献
16.
A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well‐defined relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
17.
Hailong Guo & Xu Yang 《Communications In Computational Physics》2022,31(4):1162-1179
This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities
of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to
train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo
method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method. 相似文献
18.
Rongfang Gong Xiaoliang Cheng & Weimin Han 《Communications In Computational Physics》2016,19(1):226-250
In this paper, we introduce and study a new method for solving inverse
source problems, through a working model that arises in bioluminescence tomography
(BLT). In the BLT problem, one constructs quantitatively the bioluminescence source
distribution inside a small animal from optical signals detected on the animal's body
surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization
is used to obtain stable approximate solutions. In conventional Tikhonov
regularization, it is crucial to choose a proper regularization parameter for trade off
between the accuracy and stability of approximate solutions. The new method is
based on a combination of the boundary condition and the boundary measurement
in a parameter-dependent single complex Robin boundary condition, followed by the
Tikhonov regularization. By properly adjusting the parameter in the Robin boundary
condition, we achieve two important properties for our new method: first, the regularized
solutions are uniformly stable with respect to the regularization parameter
so that the regularization parameter can be chosen based solely on the consideration
of the solution accuracy; second, the convergence order of the regularized solutions
reaches one with respect to the noise level. Then, the finite element method is used to
compute numerical solutions and a new finite element error estimate is derived for discrete
solutions. These results improve related results found in the existing literature.
Several numerical examples are provided to illustrate the theoretical results. 相似文献
19.
A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient 下载免费PDF全文
This paper is to present a finite volume element (FVE) method based on the
bilinear immersed finite element (IFE) for solving the boundary value problems of the
diffusion equation with a discontinuous coefficient (interface problem). This method
possesses the usual FVE method's local conservation property and can use a structured
mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient
has discontinuity along piecewise smooth nontrivial curves. Numerical examples are
provided to demonstrate features of this method. In particular, this method can produce
a numerical solution to an interface problem with the usual O(h2) (in L2 norm)
and O(h) (in H1 norm) convergence rates. 相似文献
20.
Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis 下载免费PDF全文
David Kamensky John A. Evans & Ming-Chen Hsu 《Communications In Computational Physics》2015,18(4):1147-1180
The purpose of this study is to enhance the stability properties of our recently-developed
numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A.
Aggarwal, Y. Bazilevs, M. S. Sacks, T. J. R. Hughes, "An immersogeometric variational
framework for fluid-structure interaction: Application to bioprosthetic heart valves",
Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based
representations of shell structures into unsteady viscous incompressible flows.
In the cited work, we formulated the fluid-structure interaction (FSI) problem using
an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian
as a set of collocated constraints, at quadrature points of the surface integration
rule for the immersed interface. Because the density of quadrature points is not
controlled relative to the fluid discretization, the resulting semi-discrete problem may
be over-constrained. Semi-implicit time integration circumvents this difficulty in the
fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems
that approach steady solutions, though, we find that spatially-oscillating modes
of the Lagrange multiplier field can grow over time. In the present work, we stabilize
the semi-implicit integration scheme to prevent potential divergence of the multiplier
field as time goes to infinity. This stabilized time integration may also be applied in
pseudo-time within each time step, giving rise to a fully implicit solution method. We
discuss the theoretical implications of this stabilization scheme for several simplified
model problems, then demonstrate its practical efficacy through numerical examples. 相似文献