Numerical analysis seminars
Spring semester 2011
Thursday 30 June 2011, 14:15–15:00, room 1329
Speaker: Klas Modin, Massey University
Title: Generalised Euler equations: volume-preserving diffeomorphisms and conformal embeddings
In this second talk on generalised Euler equations I will first demonstrate the standard derivation of the Euler equations for the volume preserving diffeomorphisms of a compact Riemannian manifold. Thereafter I will present some recent results on a new "Euler-like" equation corresponding to a reduced geodesic equation for the infinite dimensional Lie-Fréchet manifold of planar conformal embeddings of the disk. Both equations admit particle and sheet solutions, corresponding to totally geodesic submanifolds. From a numerical point of view, these solutions are interesting as they acknowledge a method for structure preserving spatial discretisation.
Thursday 23 June 2011, 14:15–15:00, room 1329
Speaker: Klas Modin, Massey University
Title: The geodesic approach to diffeomorphic template matching, I
The field of geodesics on Lie groups started in the seminal work by Vladimir Arnold, who in 1966 produced a novel interpretation of the Euler equations in fluid dynamics as the geodesic equation on the group of volume preserving diffeomorphisms with respect to an invariant metric. Arnold realised that the same approach could be used to derive equations on any Lie group of finite or infinite dimension. Since then the research on generalised Euler equations has grown explosively. Many well-known equations of mathematical physics are now realised to be generalised Euler equations.
I will talk about how the framework of generalised Euler equations can be applied to diffeomorphic shape and image matching, where one shape or image is warped into another by a smooth transformation. In particular, some recent results on the generalised Euler equation for conformal transformations are given.
Tuesday June 14th, 13:15, room 1329
Speaker: Yuto Miatake
Title: Structure preserving finite difference schemes for the Ostrovsky equation
Abstract: We consider structure preserving integration of the Ostrovsky equation. This equation has two associated invariants, the norm and energy functions, and recently Yaguchi–Matsuo–Sugihara(2010) have proposed several finite difference schemes preserving the invariants. In this talk, we first propose yet other conservative finite difference schemes, and confirm that they are more advantageous than the existing schemes. Next we find a multi-symplectic formulation of the equation, and derive multi-symplectic finite difference schemes based on the expression.
Wednesday 8th of june 2011, 14:15-15:00, room 1329
Speaker: Andreas Asheim
Title: Complex Gaussian quadrature for oscillatory integral transforms
The classical theory of Gaussian quadrature assumes a positive weight function. This guarantees existence and uniqueness of the orthogonal polynomials whose zeros are the nodes of the Gaussian quadrature. Moreover we are guaranteed nodes that are contained in the interval of integration, as well as positive quadrature weights. This implies that the rule is stable. Still, in many cases we can in fact construct orthogonal polynomials with respect to oscillatory weights. We shall show that these rules will be well suited for evaluating many oscillatory integral transforms with large arguments since they exhibit high asymptotic order. The notion of quadrature nodes contained in the interiour of the interval of integration will have to be abandoned, but in many cases we can show existence of the rules and postitive quadrature weight. For the Fourier oscillator we will get the so-called numerical method of steepest descent. However, when considering more general kernels like e.g. $J_\nu$, the kernel of the Hankel transform, more exotic quadrature rules emerge.
Wednesday 18th of May 2011, 11:15-12:00, room F2 (Gamle Fysikk)
Speaker: Prof. Richard Tsai
Title: Multiscale modeling and computation of highly oscillatory dynamical systems
Oscillatory dynamical systems can be found in many applications ranging from celestial mechanics to molecular dynamics. Typically, in such applications, one is interested in certain effective properties of the system that can only be extracted from the solutions in a time interval much larger than the period of the smallest oscillations. Computationally, resolving fast oscillations accurately in such long time intervals is very challenging. However, for systems with scale separation in the oscillations' frequencies, multiscale algorithms can be designed such that the effective behavior of the system is accurately modeled and computed without resolving the fast scales in the entire time interval. We discuss several issues arising in designing multiscale algorithms that aim at achieving this goal.
Thursday 28th of April 2011, 14:15-15:00, room 1329
Speaker: Prof. Bojan Orel
Title: Approximate solution of BVP with nonperiodic Fourier series
The talk will be on the approximate solutions of nonperiodic boundary value problems (BDFs) by truncated trigonometric series. The method is based on Huybrechts' technique for approximating nonperiodic functions by trigonometric series.
In the first part of the talk we'll describe the efficient construction of the half-range Chebyshev polynomials of the first and second kind and their role in approximating functions with trigonometric series, as well as efficient manipulations with these series (computing derivatives and products of two such series). In the second part applications of these techniques to BVP in ODEs will be considered.
Thursday 7th of April 2011, 15:15-16:00, room 1329
Speaker: Dr. Lina Song
La Trobe University, Melbourne, Australia.
Title:Nonlinear stability analysis of symplectic integrators
In this talk, I will introduce the stability of symplectic methods, especially the nonlinear stability. We induce the nonlinear stability of symplectic integrators by simulating a kind of Hamiltonian systems with a homoclinic orbit. The nonlinear stability is tested by a nonlinear Hamiltonian system of one degree of freedom with one homoclinic orbit. The nonlinear stability set is given by calculating the critical value of step size which generates a homoclinic orbit through the preserved hyperbolic equilibrium of the truncated modified equation of the considered symplectic algorithm. We find primary relations between the linear stability and the nonlinear stability of symplectic algorithms.
Wednesday 30th of March 2011, 14:15-15:00, room 1329
Speaker: Dr. Lina Song
La Trobe University, Melbourne, Australia.
Title:Poisson integrators for Lie-Poisson structures on R^3
Abstract: This talk is concerned with the study of Poisson integrators. We are interested in Lie-Poisson systems on R^3. First, we focus on Poisson integrators for constant Poisson systems and the transformations used for transforming Lie-Poisson structures to constant Poisson structures. Then, we construct local Poisson integrators for Lie-Poisson systems on R^3. At last, we present the results of numerical experiments for two Lie-Poisson systems and compare our Poisson integrators with other known methods.
Thursday 17th March, 14:15-15:00, room 1329
Speaker: Professor Christian Thaulow
Dept Engineering Design and Materials, NTNU
Title: Atomistic Modeling of Materials Failure
Abstract:
Whether a material is ductile or brittle depends on the competition of intrinsic material parameters (such as the energy required creating new surfaces versus the energy required to initiate shearing of the lattice to form and move a dislocation). The type of mechanical failure response will be controlled by the temperature and deformation rate. Experimental studies of single crystal silicon with pre-cracks have shown that at temperatures below about 850K the material tends to be extremely brittle, while it exhibits ductile behavior above this temperature. Several explanations have been proposed; however, thus far no direct atomistic level understanding exists about the underlying process that leads to BDT in silicon. This progress has been hindered partly due to lack of atomistic models that enable the simulation of sufficiently large systems to accurately describe the fracture process. By solely raising the temperature in a series of computational experiments with otherwise identical boundary conditions, we observe a sudden change from brittle to ductile behavior between 880 K and 890 K, drastically changing the material in a very narrow ≈10 K temperature regime. Our studies elucidate a cascade of atomic mechanism that control the occurrence of the BDT. We find that at elevated temperatures, the formation of a small amorphous region at an atomically sharp crack tip creates a cleavage ledge at the crack tip, inducing local mode II (shear) stresses at the crack tip, which in turns leads to dislocation emission. Our results provide a fundamental understanding of the link between stress the crack tip geometry, associated structural changes under temperature variations, and the overall mechanical behavior of a solid. Our simulations provides important insight into the atomistic-level mechanism of the brittle-to-ductile transition in silicon, with relevance for other materials that undergo BDT. The presentation will also include ongoing research on BDT for bcc-Fe, including machining of pillars and fracture mechanics specimens with Focused Ion Beam and subsequent testing with nanoindenter equipment
Thursday 24th February, 14:15-15:00, room 1329
Speaker: Jens Lohne Eftang
Title: The Empirical Interpolation Method
Abstract:
The empirical interpolation method (EIM) is a method for approximation of parameter dependent functions. The EIM was originally developed as a tool within the reduced basis framework for parametrized partial differential equations. When the parameter dependence is "non-affine," the approximation provided by the EIM accommodates efficient assembly of the parametrized reduced basis operators. More generally, the EIM may provide rapid evaluation of parameter dependent integrals whenever the underlying parameter dependence is smooth.
In this talk we first introduce and discuss the EIM. We then consider two new contributions to the EIM: rigorous a posteriori estimators for the error in the EIM approximation, and some new convergence results related to approximation of derivatives with respect to the parameters.
Thursday 10th February, 14:15-15:00, room 1329
Speaker: Syvert Nørsett
Title: The Fox–Li operator
Abstract:
The Fox–Li operator is a convolution operator over a finite interval with a special highly oscillatory kernel. It plays an important role in laser engineering. However, the mathematical analysis of its spectrum is still rather incomplete. In the present paper we show how standard Wiener–Hopf theory can be used to obtain insight into the behaviour of the singular values of the Fox–Li operator. In addition, several approximations to the spectrum of the Fox–Li operator are discussed and results on the singular values and eigenvalues of certain related operators are derived.
Fall semester 2010
Tuesday 7th December, 14:15-15:00, room 1329
Speaker: Farid Bozorgnia
Title: Numerical Approximations for Minimization problems and related Free Boundary Problems.
Abstract: In this talk, numerical approximations for some class of minimization problems are presented. The first class is a variational problem of the Spatial Segregation of Reaction-diffusion Systems. For this problem, based on finite difference method, we use quantitative properties of the solution and free boundaries. In order to accelerate our scheme the idea of multi-grid method is implemented.
The second minimization problem is called multi phase problem which can be considered as generalization of well know one phase and two phase Obstacle problems. The last minimization problem called m-membrane problem. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem.
Tuesday 9th Novber, 14:15-15:00, room 1329
Speaker: Olivier Verdier
Title: Generalized inf-sup conditions
In order to solve a saddle point problem, like the stationary Stokes equation, one has to make sure that the chosen functional spaces fulfill a condition called the "inf-sup" condition. I will show how this may be generalized to general, infinite dimenional, linear problems. For example elastostatics, and various "mixed" formulations of the Poisson problem.
The essential idea is that one may progressively reduce the system, with respect to another operator (usually given by the time-dependent part, if any). Those reductions occur in two flavours: "observation" and "control" type. In order for each reduced system to be equivalent to the original system, some compatibility conditions must be fulfilled. In the finite dimensional case, these conditions reduce to the "regular pencil" conditions, and the number of reductions is the index of the system. In the saddle point problem case, two reductions, one of each kind, are necessary, and the compatibility conditions exactly reduce to the standard inf-sup condition.
The most striking result is that the order of the reduction sequence does not matter if the inf-sup conditions are fulfilled. This fact is well known in the finite dimensional case, but comes as a surprise in the infinite dimensional case.
Tuesday 26th October, 14:15-15:00, room 1329
Speaker: Arne Morten Kvarving
Title: Bènard cells and pattern formation: A numerical investigation
Bènard-Marangoni convection is a rich and intriguing flow problem. When a temperature gradient is induced across a thin fluid layer, convection in the flow occurs due to surface-tension effects. The flow organizes itself in predominantly hexagonal convection cells with upflow at the center and downflow at the edges. While this is the general rule, anomalies in the patterns have been observed in experimental studies, yet d no good explanation for their appearance have been given. We have developed a fast, specialized solver enabling us to perform a statistical study of pattern formation, in an attempt to illuminate the cause of these anomalies. In this talk, I will describe this solver and its implementation on combined shared/distributed memory machines. Numerical results and a discussion of these will be given in the end.
Tuesday 12th October, 14:15-15:00, room 1329
Speaker: Andreas Asheim
Title: A problem from acoustics and numerical uniform asymptotics
Highly oscillatory quadrature methods show promising performance when applied to a range of real-life problems. These are methods for which the error can, under reasonable circumstances, be controlled, and they exhibit asymptotic accuracy. Asymptotic accuracy means better accuracy the higher the frequency of oscillation, a property that is inherited from the asymptotics of the quadrature problem. It turns out, however, that such quadrature methods inherit some bad traits from the asymptotics as well. In classical asymptotics this is the background for uniform asymptotic approximations. In this talk I will present some examples of this, starting from a computational problem in acoustics, and then report a bit on the ongoing search for what we'd like to call numerical uniform asymptotic methods.
Tuesday 14th September, 14:15-15:00, room 1329
Speaker: Jens Lohne Eftang
Title: Reduced Basis Methods for Parametrized Partial Differential Equations
Abstract:
Many engineering applications require accurate computation of outputs from solutions of partial differential equations that depend on one or more parameters. The parameters may represent geometry, material properties, applied forces, or boundary conditions. The output is typically a linear functional of the solution, such as an average of the field variable. The reduced basis (RB) method is a computational framework that provides rapid and accurate output prediction given any input parameter value from a predefined parameter domain. Moreover, the error in the output prediction can be rigorously bounded. The RB method is particularly attractive in real-time (parameter estimation, optimal control) or many-query (stochastic simulation) contexts, in which a low marginal output evalution cost is important.
In this talk we first discuss the standard RB method with emphasis on computational procedures and rigorous a posteriori error estimation. We then present a new generalization of the standard approach, the "h-p" RB method, which provides an additional (online) speedup of the RB output prediction through an automatic and optimal partition of the parameter domain.
Tuesday 31st August, 14:15-15:00, room 1329
Speaker: Takaharu Yaguchi
Title: The Discrete Variational Derivative Method Based on Discrete Differential Forms
Abstract:
As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit the property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice.
Lately, Furihata and Matsuo have developed the so-called “discrete variational derivative method” that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems.
In this talk, we will show an extension of this method to triangular meshes. This extension is achieved by combination of this method and the theory of the discrete differential forms by Bochev and Hyman.
Spring semester 2010
Thursday 27th May, 14:15-15:00, room 1329
Speaker: Natalia Ramzina
Title: Valuation of Barrier Option by Simulation
Abstract:
Barrier options are one of the most popular forms of path-dependent options. They provide the appropriate hedge in a number of situations and are, at the same time, less expensive than the corresponding standard options. In this thesis, the Monte Carlo approach to pricing the barrier options was implemented. Applying this method to the options with one barrier, we analyzed results of different examples with the various number of barrier level values. The convergence of the Monte Carlo method was obtained and the numerical results agree with the theoretical claims. Finding a price of a double barrier option is the major goal. We give a description of the program in the case of an option with two barriers, when the payoff is determined by user. In addition, results of tests with different choices of the barriers are presented.
Monday 10th May, 14:15-15:00, room 1329
Speaker: Kjetil Andre Johannessen
Title "Computational geometry: NURBS and T-splines""
Abstract
Traditionally non-uniform rational B-splines (NURBS) have been used almost exclusively in the design and geometry community. However, with the work on isogeometry introduced by Hughes et. al. the technology have received huge amount of interest from the numerics and analysis community as well. I will give an introduction into state-of-the-art spline representation and also the influence this will have when used in a finite element setting.
Monday 03 May, 14:15-15:00, room 1329
Speaker: Moody Chu, North Carolina State University, Raleigh, North Carolina
Title "Semi-definite Programming Techniques for Structured Quadratic Inverse Eigenvalue Problems"
Abstract
The quadratic model: Mx¨ + Cx˙ + Kx = f (t), where x 2 Rn and M, C, K 2 Rn×n usually are structured, arises in many important applications. Specifications of the underlying physical system are embedded in the matrix coefficients M, C and K while the resulting bearing of the system usually can be interpreted via its eigenvalues and eigenvectors. The process of analyzing and deriving the spectral information and, hence, inducing the dynamical behavior of a system from a priori known physical parameters such as mass, length, elasticity, inductance, capacitance, and so on is referred to as a direct problem. The inverse problem, in contrast, is to validate, determine, or estimate the parameters of the system according to its observed or expected behavior. The concern in the direct problem is to express the behavior in terms of the parameters whereas in the inverse problem the concern is to express the parameters in term of the behavior. The inverse problem is just as important as the direct problem in applications. The notion of quadratic inverse eigenvalue problems (QIEPs) is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints suits well to many engineering applications. The main thrust in this work must take into account two critical constraints arisen in practice: (a) In a large or complicated physical system, not all eigeninformation is retrievable and, for high complexity systems, not all eigeninformation retrieved is reliable. In the field of model updating, sometimes certain eigenvectors are required to satisfy some specific conditions. In all, only a few measured eigenvectors and eigenvalues are available. It is more sensible and practically implementable to employ only the available, reliable, yet limited eigenpair information in the reconstruction or updating process. (b) Depending on the physical applications, the coefficient matrices (M, C, K) often inherit some common structures such as symmetry or positive definiteness. The most challenging structure, nonetheless, comes from the inner-connectivity of elements in the original physical configuration. The connectivity constraint mandates a certain zero patterns or algebraic relationships among the entries of (M, C, K). What is even more compounding is that these patterns and relationships vary from problem to problem. For feasibility, it is necessary that the QIEP takes this connectivity into account. Indeed, merely solving the QIEP subject to the structural constraint is not enough. We must insist that the recovered parameters remain nonnegative for physical realization. Issues of solvability, computability, sensitivity, and feasibility concerning the construction and updating of the quadratic pencil (M, C, K) are highly problem dependent. Thus far, the QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. In this talk, the speaker will describes an innovative application of semi-definite techniques (SDP) to the QIEPs. Of notably interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.
Monday 26 April, 14:15-15:00, room 1329
Speaker: Anne Kværnø, NTNU.
Title: Stochastic B-series
Abstract: We employ a unifying approach for the construction of stochastic B-series which is valid both for Itˆo- and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of e.g. Runge–Kutta methods. Moreover, the analytical techniques applied in this paper may be of use in many other similar contexts.
Monday 12 April, 14:15-15:00, room 1329
Speaker: Daniel Simpson, NTNU.
Title: Matrix function methods for fractional partial differential equations
Abstract: Space-fractional partial differential equations occur when modeling processes undergoing anomalous diffusion. These model involve fractional powers of elliptic partial differential operators and, as such, are quite difficult to solve numerically. In particular, as these operators are non-local, any reasonable discretization will result dense matrices. In this talk I will look at matrix function methods that allow the solution of these problems using only sparse matrix operations. This framework also allows for easy modification of existing numerical code.
Monday 22 March, 14:15-15:00, room 1329
Speaker: Mohamed El Ghami, University of Bergen
Title "Efficient algorithms for linear and semidefinite optimization problems based on kernel functions"
Monday 15 March, 14:15-15:00, room 1329
Speaker: Olivier Verdier, NTNU.
Title "Python for Scientific Computing".
Abstract
Python along with the relevant scientific libraries can be used for scientific computations. This solution is a powerful and open source alternative to Matlab.
I will present Python, its most useful scientific libraries, and I will discuss its use in research and teaching.
Monday 8th of March, 14:15-15:00, room 1329
Speaker: Øystein Tråsdahl, NTNU.
Title " High order polynomial approximations in deformed domains: optimal point distributions ".
Abstract
The motivation behind this work is the numerical solution of partial differential equations in deformed domains based on high order polynomial approximations. Within each subdomain the pertinent field variables are approximated as high order polynomials over an associated undeformed reference domain.
In the context of a fixed and known geometry, we discuss how to best define the mapping between the reference domain and an isoparametric approximation of the physical domain, in particular, how to choose the points along the domain boundary. Numerical results are presented for different strategies, and comparisons are made based on different metrics. Some of the results are also of interest in the context of constructing high order interpolants of a general function.
Monday 22nd of February, 14:15-15:00, room 1329
Speaker: Håkon Marthinsen, NTNU.
Title "Lie group methods and canonical coordinates of the second kind".
Abstract I will start the talk with an introduction to Lie group integrators, and talk about why these are especially useful in cases where we have some kind of conservation. Then I will discuss the transition from Lie groups to Lie algebras and why the choice of coordinate mapping is important. Next, I will present canonical coordinates of the second kind, and show how we can utilise a special ordering of the Lie algebra basis (called an admissible ordered basis) so that the number of computations is low. Finally, if time permits, I will present some numerical results. This talk is based on the paper "Integration methods based on canonical coordinates of the second kind" by B. Owren and A. Marthinsen.
Monday 15th of February, 14:15-15:00, room 1329
Speaker: Bawfeh K. Kometa, NTNU.
Title " Semi-Lagrangian exponential integrators for index-2 differential algebraic systems ".
Abstract
Implicit-explicit (IMEX) multistep methods are very useful for the time discretization of convection diffusion PDE problems such as the Burgers equations and also the incompressible Navier-Stokes equations. Semi-discretization in space of the latter typically gives rise to an index-2 differential- algebraic (DAE) system of equations. Runge-Kutta (RK) type methods have been considered for the time discretization of such DAE systems. Some of the RK-type methods achieve high order of convergence with comparatively little storage requirements and have good stability properties. However, due to their implicit nature, they generally have a drawback over the IMEX multistep methods in terms of computational costs per step. In this talk I will propose an exponential integration method for index-2 DAEs of a special class that includes the type arising from the incompressible Navier-Stokes problem. The methods are based on the backward differentiation formulae (BDF), belong to the class of IMEX multistep methods and are unconditionally stable.
Monday 8th February, 14:15-15:00, rom 1329
Speaker: Tore Halvorsen, NTNU.
Title " Structure preserving discretizations of wave equations from theoretical physics ".
Abstract
In this talk I will give a brief overview of my PhD. I will introduce the equations we have looked at, which are equations from theoretical physics admitting large symmetry groups, and then show how these equations can be discretized such that the symmetry is preserved at the discrete level. Finally, I will state some of the results obtained.
Monday 1st of February, 14:15-15:00, room 1329
Speaker: Morten Dahlby, NTNU.
Title "Discrete-gradient integral-preserving integrators."
Abstract In this talk I will give a short introduction to discrete-gradient integrators (DGIs). We show how one can construct fourth order DGIs which provide exact preservation of any first integral possessed by a system of ODEs. In general these schemes will be fully implicit, which means that one must solve an expensive nonlinear system in each time step. We argue that a linearly implicit version of DGIs in some cases may be more effective. While most of the talk will be in the ODE setting, I will show some examples of how one can apply the same principles to Hamiltonian PDEs.
Monday 18th January, 14:15-15:00, room 1329
Speaker: Andreas Asheim, NTNU.
Title " Numerical steepest descent with path approximations ".
Abstract Combining the classical asymptotic method of steepest descent with Gaussian quadrature yields excellent quadrature methods for oscillatory integrals. In particular, they give high asymptotic accuracy, higher than comparable truncated classical asymptotic expansions. One difficulty with these methods, when applied to certain integrals, is the computation of the paths of steepest descent. These paths are defined through non-linear equations. In this talk we shall see that a simple truncated power series expansions of the paths, which would not be allowed in classical asymptotic theory, can be used as approximations to the true paths. The asymptotic analysis of the method is rather non-trivial and give some quite unexpected results.
Joint work with D. Huybrechs, KULeuven, Belgium
Fall semester 2009
Thursday December 10th, 10:15-11:00, rom 1329 Speaker:Sigrid Leyendecker, University of Kaiserlautern (Germany). Title "Structure preserving methods in computational multibody dynamics and optimal control".
Abstract
The benefits of structure preserving algorithms for the numerical time-integration of mechanical systems, also called mechanical integrators, are widely accepted in forward dynamic simulations. On the one hand, the fidelity of the approximate solution is improved compared to standard methods by inheriting certain characteristic properties of the continuous motion to the discrete trajectory. For example, the evolution of the system's energy or momentum maps exactly represents externally applied forces, in particular they are conserved along the approximate motion of unforced systems. In addition to momentum maps, the symplectic structure underlying real dynamics is respected by certain mechanical integrators, and as a consequence, they also yield good energy behaviour. On the other hand, the preservation of these quantities stabilises the numerical integration and thus enables longterm simulation. However, in the field of motion planning and optimal control via direct methods, so far, these benefits have been less used. The dynamic optimisation method DMOCC presented here, does exploit the structure preserving properties of a variational integrator within an optimal control problem.
DMOCC is applied to simulate optimal control problems of multibody systems whose dynamics is typically governed by differential algebraic equations (DAEs). To avoid unnecessary large dimensions of the discrete equations of motion, a structure preserving discrete null space reduction is performed. Examples are presented from the fields of flexible multibody dynamics, fluid-structure interaction, biomotion, contact dynamics and bipedal gait.
Monday November 23rd, 14:15- 16:00 rom 656, 24th 14:15-16:00 rom 656, 25th, 10:00-12:00 rom 656
Speaker: Snorre Christiansen UiO. Title: "Finite elements and differential forms- Part I, II and III".
Abstract
We give an overview of the theory of approximation of wave equations by Galerkin methods. It treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation. We emphasize differential operators, such as the curl, which have large kernels and use L2 stable interpolators preserving them. The second part is devoted to a framework for the construction of finite element spaces of differential forms on cellular complexes. Material on homological and tensor algebra as well as differential and discrete geometry is included. Whitney forms, their duals, their high order versions, their tensor products and their hp-versions all fit.
Contents of the first lecture: Maxwell's equations, Galerkin method, finite element spaces for div- and curl-conformal vector fields in i R^3.
Contents of the second lecture: Differential forms, simplicial and cellular decompositions, Whitney forms. De Rham's map and theorem.
Contents of the third lecture: Finite element space of differential forms. Tensor product, higher order elements. Minimal spaces.
Monday November 9th, 14:00, rom 1329.
Speaker: Olivier Verdier, NTNU. Title: Normal form for pairs of matrices and applications
I will discuss a new way of deriving the normal form of a pair of matrices. In particular it allows to prove the Kronecker decomposition theorem and the regular pencil theorem in a simple manner. It may also be used to characterize the observability/controllability indices. It also has consequences on the invariants of differential algebraic equations. In particular, it brings together the notions of "strangeness" and "tractability" indices. I will discuss those invariants (the index is just one of them) for some formulations of linearised mechanics and electrical circuits. If time allows, I will present an algorithm to compute those invariants.
Monday October 26th, 14:00, rom 1329.
Speaker: Tormod Bjøntegaard, NTNU. Title: "Accurate interface tracking for arbitrary Lagrangian-Eulerian schemes".
Abstract The ability to accurately follow time-dependent surfaces is very important in many areas of computational science and engineering. Computational methods for solving such problems is typically classified into two categories: explicit interface tracking and implicit interface tracking. We will here focus on the former class.
We consider problems formulated in the arbitrary Lagrangian-Eulerian (ALE) framework. In this framework a typical approach is to enforce a kinematic condition which dictates that the normal component of the domain velocity is equal to the normal component of the fluid velocity. For a discretized problem the surface is represented by a grid, and the kinematic condition is typically enforced at the individual grid nodes. For the tangential component of this grid velocity there is however significant freedom, and it is through the choice of this component that it is possible to ensure that a good grid-quality is maintained during the simulation.
In this talk we present a new method which is able to both accurately track the interface {\em and} maintain a good distribution of grid points. We consider an initial interface which is ``immersed'' in a known velocity field and use the new method and other common strategies to follow this interface. We give numerical results which compare different strategies, and demonstrate first, second and third order temporal accuracy when tracking deformed interfaces in two and three space dimensions.