Partial Differential Algebraic Equations Research Articles

Model hierarchy-based optimal control of radiative heat transfer

We present a model hierarchy multilevel optimisation approach to solve an optimal boundary control problem in glass manufacturing. The process is modelled by radiative heat transfer and formulated as an optimal control problem restricted by partial differential algebraic equations (PDAEs) and additional control constraints. We consider a sequence of model approximations given by space-time dependent non-linear PDAEs of ascending accuracy. The different models allow for a model hierarchy-based optimisation approach, where the models are shifted automatically as the optimisation proceeds. We present a realisation of a multilevel generalised SQP method within the fully space-time adaptive optimisation environment Kardos using linearly implicit methods of Rosenbrock type and multilevel finite elements. We apply the optimal control algorithm to a glass cooling problem and present numerical experiments for the model hierarchy-based approach in two spatial dimensions and for a fully space-time adaptive optimisation in three spatial dimensions.

Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter.

Grain boundary migration and grooving in thin 3-D systems

We employ numerical simulations and experimental methods to study grain boundary migration in idealized 3-D systems of three grains in a thin film, focusing on hole formation and annihilation of small grains. The initial structure is taken to be columnar, and isotropy is assumed for simplicity. The grain boundaries and the external surfaces of the three-grain system are assumed to be governed by mean curvature motion and surface diffusion, respectively. Along the thermal grooves, where the grain boundaries and the exterior surfaces couple, balance of mechanical forces, continuity of the surface chemical potential, and balance of mass flux dictate the boundary conditions. Using a parametric description for the evolving surfaces yields partial differential algebraic equations which are solved with finite-difference schemes on staggered grids. The evolution of the three-grain system results in hole formation or annihilation of the smallest grain, depending on the relative size of the smallest grain. Shortly prior to annihilation, the exterior surface of the smallest grains may invert, partially losing its convexity, with annihilation being accompanied by accelerated pitting rates. Similar features were observed experimentally.

Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

On finite element method–flux corrected transport stabilization for advection–diffusion problems in a partial differential–algebraic framework

An extension of the finite element method–flux corrected transport stabilization for hyperbolic problems in the context of partial differential–algebraic equations is proposed. Given a local extremum diminishing property of the spatial discretization, the positivity preservation of the one-step θ-scheme when applied to the time integration of the resulting differential–algebraic equation is shown, under a mild restriction on the time step size. As a crucial tool in the analysis, the Drazin inverse and the corresponding Drazin ordinary differential equation are explicitly derived. Numerical results are presented for non-constant and time-dependent boundary conditions in one space dimension and for a two-dimensional advection problem with a sinusoidal inflow boundary condition and the advection proceeding skew to the mesh.

Steric effects on ion dynamics near charged electrodes

We present a modified Poisson–Nernst–Planck (mPNP) model to include steric effects on ion dynamics near charged electrodes. This contribution appears directly on chemical potential of each ion in solution as a non-electrostatic term. The model consists in a partial-differential-algebraic equation (PDAE) system, which after spatial discretization was solved by the well-established DASSLC code, implemented in EMSO simulator. A careful dimensional analysis was carried out to turn the PDAE system dimensionless with normalized spatial domain. Size correlation (SC) effects for equally sized ions avoid unphysical charge densities near the electrodes (especially at high voltages), and also give some insights on double layer formation. Time evolution of free charge density profiles show a diffusion controlled dynamics with a very fast response (t∼10−8s). The methodology proposed here allows the inclusion of new contributions on PNP model framework for ion dynamics, as well as to solve both stationary and dynamical ion distributions, taking advantage of the EMSO software simulator features.

On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations

A boundary value problem for linear partial differential algebraic systems of equations with multiple characteristic curves is examined. It is assumed that the pencil of matrix functions associated with this system is smoothly equivalent to a special canonic form. The spline collocation is used to construct for this problem a difference scheme of an arbitrary approximation order with respect to each independent variable. Sufficient conditions are found for this scheme to be absolutely stable.

On the convergence rate of dynamic iteration for coupled problems with multiple subsystems

In multiphysical modeling coupled problems naturally occur. Each subproblem is commonly represented by a system of partial differential-algebraic equations. Applying the method of lines, this results in coupled differential-algebraic equations (DAEs). Dynamic iteration with windowing is a standard technique for the transient simulation of such systems. In contrast to the dynamic iteration of systems of ordinary differential equations, convergence for DAEs cannot be generally guaranteed unless some contraction condition is fulfilled. In the case of convergence, it is a linear one.In this paper, we quantify the convergence rate, i.e., the slope of the contraction, in terms of the window size. We investigate the convergence rate with respect to the coupling structure for DAE and ODE systems and also for two and more subsystems. We find higher rates (for certain coupling structures) than known before (that is, linear in the window size) and give sharp estimates for the rate. Furthermore it is revealed how the rate depends on the number of subsystems.

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

AbstractIn this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi‐interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parametercare given. With the use of the appropriate collocation points, the method for PDAEs with index‐2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index‐2 or an impulse solution with index‐2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95–119, 2014

A numerical method for solving partial differential algebraic equations

Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.

Electrochemical Model Based Observer Design for a Lithium-Ion Battery

Batteries are the key technology for enabling further mobile electrification and energy storage. Accurate prediction of the state of the battery is needed not only for safety reasons, but also for better utilization of the battery. In this work we present a state estimation strategy for a detailed electrochemical model of a lithium-ion battery. The benefit of using a detailed model is the additional information obtained about the battery, such as accurate estimates of the internal temperature, the state of charge within the individual electrodes, overpotential, concentration and current distribution across the electrodes, which can be utilized for safety and optimal operation. Based on physical insight, we propose an output error injection observer based on a reduced set of partial differential-algebraic equations. This reduced model has a less complex structure, while it still captures the main dynamics. The observer is extensively studied in simulations and validated in experiments for actual electric-vehicle drive cycles. Experimental results show the observer to be robust with respect to unmodeled dynamics as well as to noisy and biased voltage and current measurements. The available state estimates can be used for monitoring purposes or incorporated into a model based controller to improve the performance of the battery while guaranteeing safe operation.

The numerical solution of partial differential-algebraic equations

In this paper, a numerical solution of partial differential-algebraic equations (PDAEs) is considered by multivariate Pade approximations. We applied this method to an example. First, PDAE has been converted to power series by two-dimensional differential transformation, and then the numerical solution of the equation was put into a multivariate Pade series form. Thus, we obtained the numerical solution of PDAEs.

A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method

We have considered linear partial differential algebraic equations (LPDAEs) of the form , which has at least one singular matrix of . We have first introduced a uniform differential time index and a differential space index. The initial conditions and boundary conditions of the given system cannot be prescribed for all components of the solution vector here. To overcome this, we introduced these indexes. Furthermore, differential transform method has been given to solve LPDAEs. We have applied this method to a test problem, and numerical solution of the problem has been compared with analytical solution.

Multilevel optimization for PDAE‐constrained optimal control problems with pointwise constraints on control and state

AbstractWe have developed a fully adaptive optimization environment suitable to solve complex optimal control problems restricted by partial differential algebraic equations (PDAEs) and pointwise constraints on the control [1, 2]. This contribution is devoted to the inclusion of pointwise constraints on the state within the optimization environment. To this end we first give a brief introduction into the architecture of the environment and the inclusion of pointwise constraints on the state by Moreau‐Yosida regularization. Then, we test the new tool by applying it to an optimal boundary control problem for the cooling of hot glass down to room temperature, modeled by radiative heat transfer and semi‐transparent boundary conditions. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A parallel, fully coupled, fully implicit solution to reactive transport in porous media using the preconditioned Jacobian-Free Newton-Krylov Method

Modeling large multicomponent reactive transport systems in porous media is particularly challenging when the governing partial differential algebraic equations (PDAEs) are highly nonlinear and tightly coupled due to complex nonlinear reactions and strong solution-media interactions. Here we present a preconditioned Jacobian-Free Newton-Krylov (JFNK) solution approach to solve the governing PDAEs in a fully coupled and fully implicit manner. A well-known advantage of the JFNK method is that it does not require explicitly computing and storing the Jacobian matrix during Newton nonlinear iterations. Our approach further enhances the JFNK method by utilizing physics-based, block preconditioning and a multigrid algorithm for efficient inversion of the preconditioner. This preconditioning strategy accounts for self- and optionally, cross-coupling between primary variables using diagonal and off-diagonal blocks of an approximate Jacobian, respectively. Numerical results are presented demonstrating the efficiency and massive scalability of the solution strategy for reactive transport problems involving strong solution-mineral interactions and fast kinetics. We found that the physics-based, block preconditioner significantly decreases the number of linear iterations, directly reducing computational cost; and the strongly scalable algebraic multigrid algorithm for approximate inversion of the preconditioner leads to excellent parallel scaling performance.

Optimal control of coupled multiphysics problems: Guidelines for real‐life applications demonstrated for a complex fuel cell model

AbstractIn practice one often is confronted with the simulation and optimization of dynamical processes of multiphysics systems depending on time and space. This paper shall serve as a guidebook for practioneers who wants to proceed from simulation to optimization. We are specially concerned in this paper with optimal control problems involving partial differential equations. After providing an insight into the theory of optimal control problems with partial differential equations, the focus of the paper lies on their numerical treatment to obtain approximate optimal solutions, or more precisely approximate candidate optimal solutions. The two basic approaches, first optimize then discretize (FOTD) and first discretize then optimize (FDTO) are first described in an abstract setting and then evaluated by means of the fuel cell example of this paper which can serve as a prototype problem for multiphysics processes. The example describes the dynamical behaviour inside a certain type of high temperature fuel cell where gas flows, heat distribution, and potential fields as well as chemical reactions are modelled by hyperbolic and parabolic partial differential as well as ordinary differential algebraic equations. The underlying problem is an optimal control problem where up to 28 instationary partial differential algebraic equations in one, respectively two spatial dimensions are involved. In the conclusion pros and cons of the aforementioned approaches are discussed. Hereby, not only the numerical efficiency but also the investigation of human resources are balanced against each other (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new boundary interpolation technique for parameterisation of planar surfaces with four arbitrary boundary curves

This article is concerned with construction of planar parametric surfaces enclosed by four arbitrary boundary curves. The available conventional methods for this purpose are algebraic interpolation method (like the Coons method) and partial differential equation method. These methods usually yield some problems. This article proposes a new method namely boundary interpolation method where the geometric properties of the boundary curves have been taken into account. A technique of simultaneous displacement of the interpolated curves from the opposite boundaries has been adopted. The geometric properties considered for displacement include weighted combination of geometric and linear displacement vectors of the two opposite generating curves. The algorithm has two adjustable parameters that control the characteristics of interpolation of one set of curves from their parents. Case studies show that this algorithm gives reasonably smooth boundary-fitted planar parametric surface compared with the other two conventional methods.

On the Tractability Index of a Class of Partial Differential-Algebraic Equations

We consider a class of nonlinear partial-differential-algebraic equations, where the nonlinearity is present only in the PDEs and in the coupling conditions, and some additional structural conditions hold. For this special class of PDAEs, we introduce and characterize simple algebraic conditions which lead to a notion of extended tractability index, and exemplify its application to coupled systems arising from microelectronics.

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thummler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.

Numerical simulation and linear well-posedness analysis for a class of three-phase boundary motion problems

We investigate the linear well-posedness for a class of three-phase boundary motion problems and perform some numerical simulations. In a typical model, three-phase boundaries evolve under certain evolution laws with specified normal velocities. The boundaries meet at a triple junction with appropriate conditions applied. A system of partial differential equations and algebraic equations (PDAE) is proposed to describe the problems. With reasonable assumptions, all problems are shown to be well-posed if all three boundaries evolve under the same evolution law. For problems involving two or more evolution laws, we show the well-posedness case by case for some examples. Numerical simulations are performed for some examples.