Showing papers on "Temporal discretization published in 2006"

Reduced-basis output bound methods for parabolic problems

Abstract: In this paper, we extend reduced-basis output bound methods developed earlier for elliptic problems, to problems described by ‘parameterized parabolic’ partial differential equations. The essential new ingredient and the novelty of this paper consist in the presence of time in the formulation and solution of the problem. First, without assuming a time discretization, a reduced-basis procedure is presented to ‘efficiently’ compute accurate approximations to the solution of the parabolic problem and ‘relevant’ outputs of interest. In addition, we develop an error estimation procedure to ‘a posteriori validate’ the accuracy of our output predictions. Second, using the discontinuous Galerkin method for the temporal discretization, the reduced-basis method and the output bound procedure are analysed for the semi-discrete case. In both cases the reduced-basis is constructed by taking ‘snapshots’ of the solution both in time and in the parameters: in that sense the method is close to Proper Orthogonal Decomposition (POD).

Comparative Analysis of Computational Methods for Limit-Cycle Oscillations

Abstract: Various methods are explored in the computation of time-periodic solutions for autonomous systems. The purpose of the work is to illuminate the capabilities and limitations of methods, including a new method developed as part of the work, not based on time integration, for the fast computation of limit-cycle oscillations (LCO). Discussion will focus on methodology, robustness, accuracy, and frequency prediction. Results for two simplified model problems are presented in which temporal discretization errors during LCO are taken to machine zero. One, a typical airfoil section with nonlinear structural coupling and the other, a nonlinear panel in high speed flow. Treatment of sharp transients during LCO is also discussed.

Optimal operation of GaN thin film epitaxy employing control vector parametrization

Abstract: An approach that links nonlinear model reduction techniques with control vector parametrization-based schemes is presented, to efficiently solve dynamic constraint optimization problems arising in the context of spatially-distributed processes governed by highly-dissipative nonlinear partial-differential equations (PDEs), utilizing standard nonlinear programming techniques The method of weighted residuals with empirical eigenfunctions (obtained via Karhunen-Loeve expansion) as basis functions is employed for spatial discretization together with control vector parametrization formulation for temporal discretization The stimulus for the earlier approach is provided by the presence of low order dominant dynamics in the case of highly dissipative parabolic PDEs Spatial discretization based on these few dominant modes (which are elegantly captured by empirical eigenfunctions) takes into account the actual spatiotemporal behavior of the PDE which cannot be captured using finite difference or finite element techniques with a small number of discretization points/elements The proposed approach is used to compute the optimal operating profile of a metallorganic vapor-phase epitaxy process for the production of GaN thin films, with the objective to minimize the spatial nonuniformity of the deposited film across the substrate surface by adequately manipulating the spatiotemporal concentration profiles of Ga and N precursors at the reactor inlet It is demonstrated that the reduced order optimization problem thus formulated using the proposed approach for nonlinear order reduction results in considerable savings of computational resources and is simultaneously accurate It is demonstrated that by optimally changing the precursor concentration across the reactor inlet it is possible to reduce the thickness nonuniformity of the deposited film from a nominal 33% to 31% © 2005 American Institute of Chemical Engineers AIChE J, 2006

Conservative solution of the Fokker–Planck equation for stochastic chemical reactions

Abstract: The Fokker–Planck equation on conservation form modeling stochastic chemical reactions is discretized by a finite volume method for low dimensional problems and advanced in time by a linear multistep method. The grid cells are refined and coarsened in blocks of the grid depending on an estimate of the spatial discretization error and the time step is chosen to satisfy a tolerance on the temporal discretization error. The solution is conserved across the block boundaries so that the total probability is constant. A similar effect is achieved by rescaling the solution. The steady state solution is determined as the eigenvector corresponding to the zero eigenvalue. The method is applied to the solution of a problem with two molecular species and the simulation of a circadian clock in a biological cell. Comparison is made with a Monte Carlo method.

Electromagnetic scattering from homogeneous dielectric bodies using time-domain integral equations

Abstract: This paper presents a stable and accurate method to compute the electromagnetic scattering from homogeneous, isotropic, and nondispersive bodies using time-domain integral equations (TDIEs). Unlike previous TDIE-based scattering work, the formulation presented here is based on the equations of Poggio, Miller, Chang, Harrington, Wu, and Tsai formulation. The method employs the higher-order divergence-conforming basis functions described by Graglia et al. and bandlimited interpolation functions to effect the spatial and temporal discretization of the integral equations, respectively. As the temporal basis functions are noncausal, an extrapolation mechanism is used to modify the noncausal system of equations to a form solvable by standard marching-on-in-time procedure. This work also explains the reason for late-time low-frequency instabilities encountered in current TDIE implementations and details a stabilization technique employed to overcome them. Numerical results demonstrate the accuracy and stability of the proposed technique.

A spectral-element time-domain solution of Maxwell's equations

Abstract: A spectral-element time-domain (SETD) method based on Gauss–Lobatto–Legendre (GLL) polynomials is presented to solve Maxwell's equations. The proposed SETD method combines the advantages of spectral accuracy with the geometric flexibility of unstructured grids. In addition, a 4th-order Runge–Kutta method for time integration provides high-order accuracy and thus reduces the temporal discretization errors. The numerical results demonstrate its spectral accuracy with the order of basis function and show the high efficiency of the proposed method due to its exponential convergence. © 2006 Wiley Periodicals, Inc. Microwave Opt Technical Lett 48: 673–680, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21440

Simulations of the Weibel instability with a High-Order Discontinuous Galerkin Particle-In-Cell Solver

Abstract: This paper presents a comparison of various particle-in-cell methodologies for numerical simulation of the Weibel instability. A convergence study with the established finite difference time domain particle-in-cell method establishes a base result. Comparison to novel particle-in-cell methods based on an implicit temporal discretization, and on a second and fifth order discontinuous Galerkin scheme (DG-PIC) provide insights into components of the DG-PIC schemes including divergence cleaning, particle weighing parameters and resolution requirements. High-order DG-PIC uses less grid points for a resolved solution making it competitive with the established finite difference method.

Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations

Abstract: The finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time-stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so-called minimum time step criteria are established for the Crank-Nicolson scheme, the Galerkin-time scheme, and the backward-difference scheme used in the temporal discretization. For the forward-difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one-dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Local defect correction for time-dependent problems

Abstract: The solutions of partial differential equations (PDEs) describing physical phenomena are often characterized by local regions of high activity, i.e., regions where spatial gradients are quite large compared to those in the rest of the domain, where the solution presents a relatively smooth behavior. Examples are encountered in many application areas; one of them is the transport of passive tracers in turbulent flow fields. A passive tracer is a diffusive contaminant in a fluid flow that is present in such low concentration that it does not influence the dynamics of the flow. A few examples of passive tracer transport from everyday life are the exhaust gases from chimneys, smoke from a cigarette, dust particles spread by the wind, etc. Understanding the influence of the main flow on the tracer material is crucial in many engineering applications, such as mixing in chemical reactors or transport of fly ashes in burners. Knowledge of the passive tracer behavior is also of fundamental importance in environmental sciences for studying dispersion of pollutants (e.g. chemical species, radioactive components) in the atmosphere or in the oceans. Since filaments of tracer material are often concentrated only in a very limited part of the computational domain, an efficient numerical solution of this type of problems requires the usage of adaptive grid techniques. In adaptive grid methods, a fine grid spacing and a relatively small time step are adopted only where the relatively large variations occur, so that the computational effort and the memory requirements are minimized. This thesis focuses on a new adaptive grid method for efficiently solving parabolic PDEs characterized by highly localized properties: Local Defect Correction (LDC). In LDC the PDE is integrated on a global uniformcoarse grid and on a local uniform fine grid; the latter is adaptively placed at each time level where the high activity in the solution occurs. At each time step, global and local solution are iteratively combined to ultimately produce a solution on the composite grid, union of global and local grid. In particular, the global approximation provides artificial boundary conditions for the local fine grid problem, while the local approximation is used to estimate the coarse grid local discretization error (or defect) and then to improve the solution globally by means of a defect correction. In the algorithm we propose, the local problem is solved not only with a smaller grid size, but also with a smaller time step than the one used for the global problem. In this way the local solution 124 Summary corrects the error in the global approximation due not only to spatial discretization, but also to temporal discretization. When the LDC iteration has come to a fixed point, it can be proved that the global and the local solution at the new time level coincide at the common points between the two grids. The method described in this thesis extends the LDC technique that was initially introduced in the literature for solving elliptic PDEs. The new LDC algorithm is tested in some concrete examples that illustrate its accuracy, efficiency and robustness. LDC is an iterative process that can be used for practical applications only if it converges sufficiently fast. In this thesis the convergence properties of the LDC method for time-dependent problems are studied in detail, both analytically and by means of numerical experiments. For both one- and two-dimensional problems we investigate the dependency of the LDC convergence rate on the discretization parameters (grid size and time step) for the coarse grid problem. In general, it is observed that LDC converges for any choice of the discretization parameters and that iteration errors are reduced by several orders of magnitude at every iteration. When the coarse and the fine grid problemare discretized applying the finite volume method, special care is needed to guarantee that a discrete conservation property holds for the LDC solution on the composite grid. In fact, if no special precautions are taken, the standard LDC method for time-dependent problems is such that fluxes across the interface between global and local grid are not necessarily in balance. In this thesis we propose a finite volume adapted LDC algorithm for parabolic PDEs. In this algorithm the defect term is adapted in such a way that, at each time step, fluxes across the interface between global and local grid are in balance at convergence of the LDC iteration. The finite volume adapted LDC algorithm is then extended to include a conservative regridding strategy. The strategy guarantees that the composite grid solution satisfies a discrete conservation law also when the local region is moved in time to follow the behavior of the solution. The LDC technique is not restricted to one level of refinement. In this thesis a multilevel LDC method for time-dependent problems is introduced. The time marching strategy is such that time integration at the finer levels can be performed with smaller time steps. Finally, the new, fast converging, conservative and multilevel LDC algorithm is applied to solve a transport problem with highly localized properties. In particular, we test the LDC method on a dipole-wall collision problem. The problem is solved both by LDC and by a Chebyshev-Fourier spectral method. When the two numerical solutions are compared, we see that the two methods yield very similar results. LDC, however, is specifically meant for solving problems whose solutions exhibit local regions of high activity, and for this reason it turns out to be a less complex algorithm than the Chebyshev-Fourier spectral method.

High order finite element solution of elastohydrodynamic lubrication problems

Abstract: In this thesis, a high-order finite element scheme, based upon the Discontinuous Galerkin (DG) method, is introduced to solve one- and two-dimensional Elastohydrodynamic Lubrication (EHL) problems (line contact and point contact). This thesis provides an introduction to elastohydrodynamic lubrication, including some history, and a description of the underlying mathematical model which is based upon a thin film approximation and a linear elastic model. Following this, typical nondimensionalizations of the equations are discussed, along with boundary conditions. Two families of problems are considered: line and point contacts. Following a review of existing numerical methods for EHL problems, a different numerical technique, known as the Discontinuous Galerkin method is described. This is motivated by the high accuracy requirement for the numerical simulation of EHL problems. This method is successfully applied to steady-state line contact problems. The free boundary is captured accurately using the moving-grid method and the penalty method respectively. Highly accurate numerical results are obtained at a low expense through the use of h-adaptivity methods based on discontinuity and high-order components respectively. Combined with the Crank-Nicolson method and other implicit schemes for the temporal discretization, highly accurate solutions are also obtained for transient line contact problems using the high order DG method for the spatial discretization. In particular, an extra pressure spike is captured, which is difficult to resolve when using low order schemes for spatial discretization. The extension of this high order DG method to the two-dimensional case (point contact) is straightforward. However, the computation in the two-dimensional case is more expensive due to the extra dimension. Hence p-multigrid is employed to improve the efficiency. Since the free boundary in the two-dimensional case is more complicated, only the penalty method is used to handle the cavitation condition. This thesis is ended with the conclusions and a discussion of future work.

Numerical analysis of a one dimensional diffusion equation for a single chamber microbial fuel cell using a linked simulation optimization (lso) technique

Abstract: Renewable energy (RE) applications are becoming a popular means of power generation within our society. Microbial fuel cells (MFCs) represent a new form of renewable energy by converting organic matter into electricity by using bacteria already present in wastewater while simultaneously treating the wastewater. Increase in MFC power density by oxygen sparging can be accomplished by aerating the MFC chamber to assure sufficient reaction rates at the cathode. This study’s numerical analysis includes the development and verification of FORTRAN computer code necessary to solve a one dimensional Diffusion Equation to model oxygen in a single chamber MFC. A rigorous verification of the effects of spatial and temporal discretization of the simulation model coupled to the LSO using a Modular InCore Nonlinear Optimization System (MINOS) FORTRAN computer code was performed. Implicit Finite Difference numerical methods were found to require a substantialy larger nodal value to that of the Galerkin Finite Element approximation nodal value discretization to obtain a similar amount of error of 0.005 from the analytical solution. The cost of oxygen sparging was found to decreased substantially by a nodal discretization of 20 to 80 nodes. A realistic oxygen sparging schedules was developed by the use of 70 to 80 nodal values in a FE linear numerical method utilizing the LSO methodology. Advanced Numerical Methods Zielke ii

Development and application of a three-dimensional orthogonal coordinate semi-implicit hydrodynamic model

Abstract: A three-dimensional, orthogonal coordinate semi-implicit hydrodynamic model in spherical coordinates that can be applied to estuarine, coastal sea, and continental shelf waters is presented. A generalized orthogonal coordinate transformation on the horizontal and a sigma coordinate transformation on the vertical, are applied to the governing equations. The governing equations are decomposed into exterior and interior modes and solved using a semi-implicit solution technique. Second-order accurate spatial and temporal discretization schemes are used on a space staggered grid. A simple flooding and drying technique is used to model the tidal flats. The model results are tested against analytical solutions for tidal circulation in an annular channel and steady residual flow generated by wind, and density differences in a rectangular channel. The predictions from the model showed very good comparison with analytical solutions for all the test cases. Three-dimensional circulation in Narragansett Bay was then studied using the developed model. The model predicted surface elevations, three-dimensional instantaneous and mean currents, salinities, and temperatures in Narragansett Bay are compared with the observations. Mean errors in the model predicted surface elevations and velocities are less than 3% and 15%, respectively. The spring and neap cycles, the shorter duration but stronger ebb dominant currents and the double flood phenomena seen in the observations are reproduced by the model. The mean estuarine currents, and the sub-tidal currents seen in the observations are also well reproduced by the model. Correlation coefficients for salinity and temperatures exceed 0.95 and 0.87, respectively.

High‐order compact numerical schemes for non‐hydrostatic free surface flows

Abstract: The development of a numerical scheme for non-hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high-order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four-stage, fourth-order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time-step and tridiagonal matrices for each derivative at each of the four stages in a time-step. Third- and fourth-order accurate boundaries for the flow variables have been developed including the top non-hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second-order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non-hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.

On finite element variational multiscale methods for incompressible turbulent flows

Abstract: Two realizations of finite element variational multiscale (VMS) methods for the simulation of incompressible turbulent flows are studied. The dierence between the two approaches consists in the way the spaces for the large scales and the resolved small scales are chosen. The paper addresses issues of the implementation of these methods, the treatment of the additional terms and equations in the temporal discretization, and the additional costs of these methods.

Numerical Performances of Explicit Cartesian Methods for Compressible Moving-Boundary Flow Problems

Abstract: Cartesian method, which in some places is mentioned as Volume-Tracking Method, is one of the popular methods used in simulating transient flow problems involving complex moving boundaries. It has the advantage of being time-saving, efficient and robust for certain types of fluid-structure interaction problems. This method is featured by a Cartesian background Euler mesh discretizing the flow domain and a moving surface cutting through it. The most critical operation of this method is treating the cells cut by the moving boundaries accurately and stably. When the Cartesian methods are applied, the temporal discretization of the governing equations of the flow can be either implicit or explicit. For simulations cases in which time-accurately capturing wave propagation or flow evolution is essential, explicit approach still plays an important role among the researchers and currently available simulation codes. The current study is focused on the numerical performances of the explicit type of Cartesian methods when applied on the compressible flow cases. The accuracy of the simulation results, stability and grid-convergence problems resulted from a moving, impermeable boundary cutting through the background mesh are addressed. Example problems include the one-dimensional piston problem and the expanding sphere flow problems. In one case the sphere expands supersonically thus a spherical shock is generated. In another case it expands at a subsonic speed and works as a monople impulse noise source. To the best knowledge of the author, the problem of expanding-sphere generated acoustic impulse has not been reported anywhere else. Simple theoretical analyses are included and results of numerical experiments are reported.Copyright © 2006 by ASME

Analysis of the response to orographic forcing of a time‐staggered semi‐Lagrangian discretization of the rotating shallow‐water equations

Abstract: A recently proposed time-staggered semi-Lagrangian discretization of the unforced rotating shallow-water equations is extended to include orographic forcing. Linear analysis shows that, as for traditional semi-implicit semi-Lagrangian schemes, it is also susceptible to spurious orographic resonance for large Courant numbers. A solution is proposed which, as shown by further linear analysis, addresses spurious orographic resonance whilst also maintaining computational stability. © Crown copyright, 2006. Royal Meteorological Society

An Optimal Control of Water Level Considering Time-Delay System

Abstract: This paper presents a numerical application of the optimal control considering time- delay system to the flood control with drainage basin. The optimal control theory is used to obtain the control values which satisfies the state equation and minimizes performance function. Feature of this study is application of time-delay system to control problems. Two kinds of control values which has time difference can be introduced. Those are quantity of water intake and release in drainage basin. These control values are correlated with each other, which can be modeled by the time-delay system. The Sakawa-Shindo method is employed for the minimization algorithm. To calculate the water flow phenomenon, the shallow water equation is employed. The Clank-Nicolson method is applied to temporal discretization and the stabilized bubble function method is applied to spatial discretization. As the numerical model, the flood flow in the Tsurumi river which locates in Yokohama city is represented.

Hanging variables in finite element time domain method with hexahedral edge elements

Abstract: Treatment of hanging variables for nested rectangular and hexahedral elements is formulated. The Galerkin-type treatment that involves the construction of intergrid boundary operator leads to symmetric mass and stiffness matrices, /spl otimes/-method is used for the temporal discretization in the finite element time domain method. The conditions for numerical stability of both implicit and explicit FETD method with hanging variables is obtained via analysis of the spectral properties of the resulting finite element matrices. Numerical examples confirm the stability of the proposed treatment hanging variables.

Error estimation and adaptivity in deformable solid dynamics

Abstract: In this dissertation we study the error due to the temporal discretization in numerical simulations of deformable solid dynamics . For that, we start from the semidiscrete equations of motion and analyze the conditions that an estimator must satisfy to compute accurate results. With them, two typical error estimators are analyzed. Later, we propose a novel methodology for the formulation of a posteriori error estimators for the most common time-stepping methods employed in solid and structural dynamics. The estimators obtained by means of this methodology are accurate even in non-smmoth problems, they can be applied both in linear and non-linear problems and can be easily implemented in _nite element codes. The proposed methodology is applied to construct error estimators for Newmark's method and for the HHT method. The good performance of these new estimators is investigated in several numerical simulations. The information given by these estimators is the starting point for developing adaptive algorithms. They change automatically the time step size to ensure that the error is smaller than a tolerance, but keeping the computational cost as low as possible. The algorithms presented are based on control theory, and di_erent techniques applied in this _eld are employed to optimize the adaptive strategies. The _nal adaptive schemes are very simple formulas which are easily implemented in existing codes. Three non-linear numerical examples are presented to validate the good behaviour of the adaptive algorithms described.

A meshless level-set scheme for interfacial flows

Abstract: This paper reports a novel meshless scheme for the numerical simulation of bubbles moving in an incompressible viscous fluid. In this paper, the motion of the bubble is modelled on the basis of the level set formulation for capturing the moving interface between the bubbles and the surrounding fluid, and the Navier-Stokes equations (with surface tension taken into account) for the ambient incompressible viscous flow. The new numerical scheme is then devised in which the Integrated Radial Basis Function Network (IRBFN) method is used for spatial discretization of the governing equations and high-order time stepping methods are used for temporal discretization. As a result, the moving interface is captured at all time as the zero contour of a smooth function (known as the level set function) without any explicit computation of the actual front location. A numerical simulation of two bubbles moving, stretching and merging in an incompressible viscous fluid is performed to demonstrate the working of the present scheme.

Flood control of urban stormwater conduit systems using the finite element method

Abstract: This paper presents a method of controlling the water levels in a conduit system by employing optimal control theory and the finite element method. A shallow-water equation is employed for the analysis of flow behaviour. Optimal control theory is utilized to obtain a control value for the target state value. The Sakawa–Shindo method is employed as a minimization technique. For the computational storage requirements, the time domain decomposition method is applied. The Crank–Nicolson method is used for temporal discretization. In addition to a method for optimally controlling water level, a method is presented for determining transversality conditions, the terminal condition of the Lagrange multiplier. Copyright © 2006 John Wiley & Sons, Ltd.

Choosing Optimal L-BGK Simulation Parameters for Time Harmonic Flows

Abstract: We explore simulations of time harmonic flows by the lattice Boltzmann method (LBM), and extend the work by Artoli et al. [1], [2]. We propose a general scheme to choose simulation parameters, under the constraints of fixed Reynolds and Womersley numbers, and with a specified simulation error. Under these constraints four free parameters must be specified: the spatial and temporal discretization, the relaxation parameter � (or related viscosity �) and the Mach number Ma. The choice of these four parameters not only influences the accuracy of the method, but also other features such as stability, convergence and execution time. Under the constraints of a fixed Reynolds and Womersley number and a specified simulation error, we choose the simulation parameters such that the execution time of the simulation is minimized. We have formulated an asymptotic error theory, where we assume three sources of error, due to spatial and temporal discretization and due to compressibility. The errors due to spatial and temporal discretizations are of first or second order, depending on the boundary conditions and the compressibility error