Spring Meeting WSC-KNS 2008

Speakers on the WSC Spring meeting 2008 (f.l.t.r.): Sander van Veldhuizen (TUD), Arthur van Dam (UU), Vijaya Ambati (UT), Willem Dijkstra (TU/e), Kamyar Malakpoor (UvA), Joost Rommes (NXP), Svetlana Dubinkina (CWI), Gerk Rozema (RuG).

Thursday April 3, 2008, the Werkgemeenschap Scientific Computing together with the Kontaktgroep Numerieke Stromingsleer is organizing a spring meeting at the Technical University Eindhoven. Eight young researchers have been selected to give a presentation on their research.
The filmhuis Zwarte Doos has been chosen as a perfect location for this meeting at the TU-Eindhoven. Here you can find more information about this location.

10.00 h.	Registration, coffee/tea, welcome
10.30 h.	Joost Rommes (NXP)	Model order reduction in industry [abstract]
11.00 h.	Svetlana Dubinkina (CWI)	Statistical mechanics of Arakawas discretizations [abstract]
11.30 h.	Coffee/tea break
11.45 h.	Kamyar Malakpoor (UvA)	Mixed hybrid finite element for compressible porous media [abstract]
12.15 h.	Sander van Veldhuizen (TUD)	On iterative solvers combined with projected Newton methods for reacting flow problems [abstract]
12.45 h.	Lunch at the lounge of the Zwarte Doos
13.45 h.	Gerk Rozema (RUG)	The quasi-simultaneous approach for partitioned systems in hemodynamics [abstract]
14.15 h.	Arthur van Dam (UU)	Moving meshes: intuition made powerful [abstract]
14.45 h.	Coffee/tea break
15.00 h.	Vijaya Ambati (UT)	Space-time variational (dis)continuous Galerkin method for free surface waves [abstract]
15.30 h.	Willem Dijkstra (TU/e)	The boundary element method: size and solvability [abstract]
16.00 h.	Slot

Organizing comittee:
Prof.dr. J.G. Verwer (CWI/KdvI), Prof.dr. A.E.P. Veldman (RuG), Prof.dr. R.M.M. Mattheij (TU/e), Drs. Margreet Nool (CWI, secretary).

Participation (including lunch) is free of charge but registration is obligatory. Please registrate before March 31 2008. Questions? Please ask: Margreet Nool

Joost Rommes
Model order reduction in industry

Physical structures and processes are modeled by dynamical systems in a wide range of application areas. During the design of very large-scale integration (VLSI) chips, for instance, dynamical systems are used to describe the low-level circuit behavior. Since these dynamical systems can become very large for modern chips, the essential simulation before production may consume hours or days of computing time. Hence there is need for efficient mathematical approaches that limit the computing time while preserving the accuracy. In this talk some realistic problems in VLSI design and simulation will be presented, together with solution methods based on model order reduction techniques.
[begin]

Svetlana Dubinkina
Statistical mechanics of Arakawas discretizations

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa (1966) which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.
[begin]

Kamyar Malakpoor
Mixed hybrid finite element for compressible porous media

It is known that many biologic tissues are composed of porous solid matrices with fluid filling the pores. The overall mechanical behavior of these tissues depends not only on the solid matrix deformation, but also on the movement of the fluid in and out of the pores during the deformation. The presence of a freely moving fluid in a porous solid implies its mechanical response. Two mechanisms are considered to play a key role in the interstitial fluid and porous solid interaction. An increase of pore pressure induces a dilation of the porous solid and compression of the rock causes a rise of pore pressure, if the fluid is prevented from escaping the pore network.

In this talk we apply the mixed finite element formulation of coupled deformation- diffusion process, within the framework of the Biot theory of poroelasticity. The variational formulation, existence, uniqueness of the solutions together with the regularity results are discussed. Numerical results by using hybridization technique for Mandel's problem will be given.
[begin]

Sander van Veldhuizen
On iterative solvers combined with projected Newton methods for reacting flow problems

The numerical modeling of laminar reacting gas flows in Chemical Vapor Deposition (CVD) processes commonly involves the solution of multi-dimensional advection-diffusion-reaction equations for a large number of reactants and intermediate species. Typically, the equations are nonlinearly and stiffly coupled through the reaction terms, which include dozens of finite rate elementary reaction steps with largely varying reaction rates. Numerical solvers in most commercial CFD codes have, in particular when time accurate solutions are required, great problems to solve such stiff systems of partial differential equations. Time accurate transient solutions are important for the study of start-up and shut-down cycli, and for the study of inherently transient CVD processes as Rapid Thermal CVD and Atomic Layer Deposition (ALD).

The main goal of this study is to develop robust and efficient numerical schemes for time-accurate simulation of CVD. Besides the implicit treatment of the reaction terms needed for stability requirements of the time integration method, conservation of non-negativity of the species concentrations is much more important, and much more restrictive towards the time-step size, than stability. Euler Backward is the only known method to be unconditionally positive, and therefore we restrict ourselves to this first order time integration method.

Since the positivity of the solution is very important, the use of Newton methods to solve the nonlinear problems is only feasible in combination with direct solvers. If iterative methods are used, it appears that the approximate solution vectors may have small negative entries. In order to prevent this, we use a projected Newton method. Projected Newton methods are widely known in optimization applications, but is not well known in the field of reacting flows. The interior linear systems are sparse, and, due to the large number of species, large. Iterative methods, and in particular preconditioned Krylov solvers, are suitable candidates to solve such systems. Choosing the best preconditioners enables us to reduce the computational time of the classical 17 species, 26 reactions chemistry model for a CVD process of silicon from silane, by a factor 20 on a single processor.
[begin]

Gerk Rozema
The quasi-simultaneous approach for partitioned systems in hemodynamics

When modeling complex systems often a modular approach is chosen. A fully simultaneous treatment of subsystems (strong coupling) requires an intensive merging of the submodels at algorithmic and software levels or the introduction of subiterations. Weak coupling methods on the other hand are cheap but prone to numerical stability problems. The quasi-simultaneous method combines the best of both worlds. Here it is presented in an unsteady setting.

In a quasi-simultaneous approach, a simple approximation of one of the submodels (interaction law) is solved simultaneously with the other submodel to prevent stability problems. Because of the simplicity of the interaction law the quasi-simultaneous approach adds only little complexity and the computational effort per time step is hardly effected.

Recent applications include a 3D compliant carotid artery bifurcation (fluid-structure interaction) and its coupling to a 0D circulation model, using a 0D approximation of the 3D flow model as interaction law. The CFD model Comflo, developed in-house, employs a finite-volume discretization of the Navier-Stokes equations and for the vessel wall a 3D implementation of the independent rings model is used.
[begin]

Arthur van Dam
Moving meshes: intuition made powerful

Moving mesh methods are one of the adaptive techniques that locally increase the accuracy of a solution. We use them to solve time dependent compressible flow problems. The points in a structured mesh are automatically moved as to align with relevant flow features, such as shock waves and rotations.

We will give an intuitive explanation of the mathematics behind mesh movement. Also we will describe some relevant techniques for applying the solver to non-trivial flow problems that contain many small flow features. Key in our solver is the combination of a self-regulating adaptivity criterion and some smartly chosen indicator quantities. The moving mesh is combined with a finite volume solver and is set up to handle systems of hyperbolic partial differential equations in 2D.
[begin]

Vijaya Ambati
Space-time variational (dis)continuous Galerkin method for free surface waves

Many water wave problems in marine and offshore engineering are studied using nonlinear free surface wave model which consists of a potential flow equation supplemented with kinematic and dynamic free surface boundary conditions. These equations can be derived from Luke's variational principle. Such a variational principle has the advantage of completely describing the physics of the problem using a single functional and follows the conservation of energy. Therefore, numerical discretization based on the variational principle may conserve discrete energy. Hence, in the present work, we consider a new space-time variational (dis)continuous Galerkin method for free surface gravity water waves.

Space-time discontinuous Galerkin (DG) finite element methods are advantageous for problems arising in fluid mechanics that have moving boundaries or interfaces in the flow domain. The main advantage of the method is the flexibility in handling the deforming grids because of the unification of space and time in the numerical formulation. Extra advantages are hp-adaptivity and suitability for parallelization. In the present water wave problem, the free surface continuously evolves in time and thus a space-time DG method is useful to handle deforming grids due to the free surface.

In the space-time variational DG method, we first establish a primal relation between the approximated velocity field and the velocity potential. Second, we derive a discrete functional analogous to the continuum functional of the free surface water wave problem using the primal relation. Subsequently, we apply Lukes variational principle on the discrete functional to obtain a discrete variational formulation for the free surface water wave problem. We obtain a consistent variational discretization by approximating the functions to be continuous in time but discontinuous in space. For linear free surface waves, the numerical discretization results into a algebraic system of equations with a compact stencil.

The numerical scheme for linear free surface waves is second order accurate for linear polynomial approximations and shows no dissipation qualitatively. The preliminary results for linear harmonic free surface waves do not show decay in the maximum amplitude of the free surface waves even when simulated for a long time. We conclude that the present numerical method is stable and accurate which can be extended to the nonlinear free surface waves. The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa (1966) which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.
[begin]

Willem Dijkstra
The boundary element method: size and solvability

Many software tools that solve mathematical problems work like a black box. The user provides the required input data and the software produces the solution to the mathematical problem. In order to be sure that the solution is correct, one needs to understand what happens behind the scene. To understand this, it is important to know the working of the numerical method on which the software is built. In this talk we show that one specific method cannot always be trusted to produce correct solutions.

Consider a boundary value problem (BVP) on a physical object with characteristic size L. Suppose that we have a numerical method that solves the BVP successfully. Now we double the size of the object to 2L. If we use the same numerical method, can it also successfully solve the BVP on this larger object?

For the boundary element method (BEM), a numerical method which solves BVPs, under certain conditions the answer to this question is negative. For instance, for the Laplace equation on a unit circle with Dirichlet boundary conditions, no unique solution is found. If we change the radius of the circle to any other arbitrary value, a unique solution can be found. Hence the solvability of the problem depends on the size of the domain. Software tools that are built upon the BEM may therefore produce incorrect solutions if the domain size is inappropriate.

The reason for this strange phenomenon lies in the field of boundary integral equations (BIEs), which form the basis for the BEM. In this talk we show that the BIE becomes singular at certain domain sizes. As a consequence the discrete version of the BIE, the linear system that is solved in the BEM, becomes ill-conditioned. We present a number of simple modifications to the standard BEM formulation to overcome these ill-conditioned systems.
[begin]