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  • 1.
    Abdulle, Assyr
    et al.
    ANMC, EPFL.
    Cohen, David
    Institut für Angewandte und Numerische Mathematik, KIT.
    Vilmart, Gilles
    ANMC, EPFL.
    Konstantinos, Zygalakis
    ANMC, EPFL.
    High weak order methods for stochastic differential equations based on modified equations2012In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 34, no 3, p. A1800-A1823Article in journal (Refereed)
    Abstract [en]

    Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (mean-square stable) stochastic problems, and implicit integrators that exactly conserve all quadratic firstintegrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

  • 2.
    Adlerborn, Björn
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Distributed one-stage Hessenberg-triangular reduction with wavefront scheduling2018In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 2, p. C157-C180Article in journal (Refereed)
    Abstract [en]

    A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented. The formulation is based on a sequential cacheblocked algorithm by K degrees agstrom et al. [BIT, 48 (2008), pp. 563 584]. A static scheduling algorithm is proposed that addresses the problem of underutilized processes caused by two-sided updates of matrix pairs based on sequences of rotations. Experiments using up to 961 processes demonstrate that the new formulation is an improvement of the state of the art and also identify factors that limit its scalability.

  • 3.
    Adlerborn, Björn
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Distributed one-stage Hessenberg-triangular reduction with wavefront scheduling2016Report (Other academic)
    Abstract [en]

    A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented. The formulation is based on a sequential cache-blocked algorithm by Kågstrom, Kressner, E.S. Quintana-Ortí, and G. Quintana-Ortí (2008). A static scheduling algorithm is proposed that addresses the problem of underutilized processes caused by two-sided updates of matrix pairs based on sequences of rotations. Experiments using up to 961 processes demonstrate that the new algorithm is an improvement of the state of the art but also identifies factors that currently limit its scalability.

  • 4.
    Adlerborn, Björn
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kressner, Daniel
    SB–MATHICSE–ANCHP, EPF Lausanne.
    PDHGEQZ user guide2015Report (Other academic)
    Abstract [en]

    Given a general matrix pair (A,B) with real entries, we provide software routines for computing a generalized Schur decomposition (S, T). The real and complex conjugate pairs of eigenvalues appear as 1×1 and 2×2 blocks, respectively, along the diagonals of (S, T) and can be reordered in any order. Typically, this functionality is used to compute orthogonal bases for a pair of deflating subspaces corresponding to a selected set of eigenvalues. The routines are written in Fortran 90 and targets distributed memory machines.

  • 5.
    Agvik, Simon
    Umeå University, Faculty of Science and Technology, Department of Physics.
    A deformable terrain model in multi-domain dynamics using elastoplastic constraints: An adaptive approach2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Achieving realistic simulations of terrain vehicles in their work environment does not only require a careful model of the vehicle itself but the vehicle's interactions with the surroundings are equally important. For off-road ground vehicles the terrain will heavily affect the behaviour of the vehicle and thus puts great demands on the terrain model.

    The purpose of this project has been to develop and evaluate a deformable terrain model, meant to be used in real-time simulations with multi-body dynamics. The proposed approach is a modification of an existing elastoplastic model based on linear elasticity theory and a capped Drucker-Prager model, using it in an adaptive way. The original model can be seen as a system of rigid bodies connected by elastoplastic constraints, representing the terrain. This project investigates if it is possible to create dynamic bodies just when it is absolutely necessary, and store information about possible deformations in a grid.

    Two methods used for transferring information between the dynamic bodies and the grid have been evaluated; an interpolating approach and a discrete approach. The test results indicate that the interpolating approach is preferable, with better stability to an equal performance cost. However, stability problems still exist that have to be solved if the model should be useful in a commercial product.

  • 6.
    Anton, Rikard
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria.
    Exponential integrators for stochastic Schrödinger equations driven by Itô noise2018In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 36, no 2, p. 276-309Article in journal (Refereed)
    Abstract [en]

    We study an explicit exponential scheme for the time discretisation of stochastic Schr¨odinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schr¨odinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

  • 7.
    Anton, Rikard
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cohen, David
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Quer-Sardanyons, Lluis
    A fully discrete approximation of the one-dimensional stochastic heat equation2018In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642Article in journal (Other academic)
    Abstract [en]

    A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in Lq(Ω)" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline-table; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">Lq(Ω)Lq(Ω)⁠, for all q⩾2" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline-table; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">q⩾2q⩾2⁠, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

  • 8.
    Araujo-Cabarcas, Juan Carlos
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Reliable hp finite element computations of scattering resonances in nano optics2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

    The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

    In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

    The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

  • 9.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Campos, Carmen
    Engström, Christian
    Roman, Jose E.
    Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structuresManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS.Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

  • 10.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Engström, Christian
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On spurious solutions in finite element approximations of resonances in open systems2017In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, no 10, p. 2385-2402Article in journal (Refereed)
    Abstract [en]

    In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.

  • 11.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Engström, Christian
    Removal of spurious solutions encountered in Helmholtz scattering resonance computations in R^dManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can therefore be applied to graded material properties as well as piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to remove spurious solutions in electromagnetic and acoustic scattering resonances computations at low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

  • 12.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Engström, Christian
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Jarlebring, Elias
    Royal Institute of Technology.
    Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map2018In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed)
    Abstract [en]

    We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

  • 13. Backman, Anders
    et al.
    Bodin, Kenneth
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Lacoursière, Claude
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Servin, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Democratizing CAE with Interactive Multiphysics Simulation and Simulators2012Conference paper (Other academic)
  • 14.
    Baek, Seung Ki
    et al.
    Umeå University, Faculty of Science and Technology, Physics.
    Bernhardsson, Sebastian
    Umeå University, Faculty of Science and Technology, Physics.
    Equilibrium solution to the lowest unique positive integer game2010In: Fluctuation and Noise Letters, ISSN 0219-4775, E-ISSN 1793-6780, Vol. 9, no 1, p. 61-68Article in journal (Refereed)
    Abstract [en]

    We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.

  • 15.
    Bartlett, Roscoe A.
    et al.
    Sandia National Laboratories.
    Van Bloemen Waanders, Bart
    Sandia National Laboratories.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Hybrid Differentiation Strategies for Simulation and Analysis of Applications in C++2008In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 35, no 1Article in journal (Refereed)
  • 16.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    A vertex-centered dual discontinuous Galerkin method2006In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 192, no 1, p. 175-181Article in journal (Refereed)
  • 17.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Approximations of very weak solutions to boundary-value problems2004In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, no 2, p. 860-877Article in journal (Refereed)
  • 18.
    Berggren, Martin
    Department of Computational and Applied Methematics, Rice University, Houston.
    Numerical solution of a flow-control problem: vorticity reduction by dynamic boundary action1998In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 19, no 3, p. 829-860Article in journal (Refereed)
  • 19.
    Berggren, Martin
    Department of Computational and Applied Mathematics, Rice University, Houston, TX.
    Solving an advection-diffusion problem on the Connection Machine1994In: Concurrency: Practice and Experience, Vol. 6, no 1, p. 55-68Article in journal (Refereed)
  • 20.
    Berggren, Martin
    FFA, the Aeronautical Research Institute of Sweden.
    The volume discharge approach to geometric conservation1999In: Computational Methods for Fluid-Structure Interaction / [ed] T. Kvamsdal, 1999Conference paper (Refereed)
  • 21.
    Berggren, Martin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Bernland, Anders
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Noreland, Daniel
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Acoustic boundary layers as boundary conditions2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 371, p. 633-650Article in journal (Refereed)
    Abstract [en]

    The linearized, compressible Navier-Stokes equations can be used to model acoustic wave propagation in the presence of viscous and thermal boundary layers. However, acoustic boundary layers are notorious for invoking prohibitively high resolution requirements on numerical solutions of the equations. We derive and present a strategy for how viscous and thermal boundary-layer effects can be represented as a boundary condition on the standard Helmholtz equation for the acoustic pressure. This boundary condition constitutes an O (delta) perturbation, where delta is the boundary-layer thickness, of the vanishing Neumann condition for the acoustic pressure associated with a lossless sound-hard wall. The approximate model is valid when the wavelength and the minimum radius of curvature of the wall is much larger than the boundary layer thickness. In the special case of sound propagation in a cylindrical duct, the model collapses to the classical Kirchhoff solution. We assess the model in the case of sound propagation through a compression driver, a kind of transducer that is commonly used to feed horn loudspeakers. Due to the presence of shallow chambers and thin slits in the device, it is crucial to include modeling of visco-thermal losses in the acoustic analysis. The transmitted power spectrum through the device calculated numerically using our model agrees well with computations using a hybrid model, where the full linearized, compressible Navier-Stokes equations are solved in the narrow regions of the device and the inviscid Helmholtz equations elsewhere. However, our model needs about two orders of magnitude less memory and computational time than the more complete model. 

  • 22.
    Berggren, Martin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Ekström, Sven-Erik
    Department of Information Technology, Uppsala University.
    Nordström, Jan
    Department of Information Technology, Uppsala University.
    A discontinuous Galerkin extension of the vertex-centered edge-based finite volume method2009In: Communications in Computational Physics, ISSN 1815-2406, Vol. 5, no 2-4, p. 456-468Article in journal (Refereed)
    Abstract [en]

    The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising high-accuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

  • 23.
    Berggren, Martin
    et al.
    Department of Computational and Applied Mathematics, Rice University, Houston TX.
    Glowinski, Roland
    Department of Mathematics, University of Houston, Houston, , TX .
    A spectral preconditioner for control problems associated with linear evolution equations1995In: East-West Journal on Numerical Mathematics, Vol. 3, no 2, p. 81-109Article in journal (Refereed)
  • 24.
    Berggren, Martin
    et al.
    FFA, the Aeronautical Research Institute of Sweden, Bromma.
    Glowinski, Roland
    Department of Mathematics, University of Houston.
    Lions, Jacques-Louis
    College de France.
    A Computational Approach to Controllability Issues for Flow-Related Models. (I): Pointwise Control of the Viscous Burgers Equation1996In: International journal of computational fluid dynamics (Print), ISSN 1061-8562, E-ISSN 1029-0257, Vol. 7, no 3, p. 237-252Article in journal (Refereed)
  • 25.
    Berggren, Martin
    et al.
    FFA, the Aeronautical Research Institute of Sweden, Bromma, Sweden.
    Glowinski, Roland
    Department of Mathematics, University of Houston, Houston.
    Lions, Jacques-Louis
    College de France, Rue d'Ulm, Paris, France .
    A computational approach to controllability issues for flow-related models. (II): Control of two-dimensional, linear advection-diffusion and Stokes models.1996In: International journal of computational fluid dynamics (Print), ISSN 1061-8562, E-ISSN 1029-0257, Vol. 6, no 4, p. 253-274Article in journal (Refereed)
  • 26.
    Berggren, Martin
    et al.
    FFA, the Aeronautical Research Institute of Sweden.
    Heinkenschloss, Matthias
    Department of Computational and Applied Mathematics, Rice University, Houston, TX.
    Parallel Solution of Optimal-Control Problems by Time-Domain Decomposition1997In: Computational Science for the 21st Century / [ed] M.O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux, and M. F. Wheeler, 1997Conference paper (Refereed)
  • 27.
    Berglund, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Numerical Simulations of Linear Stochastic Oscillators: driven by Wiener and Poisson processes2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The main component of this essay is the numerical analysis of stochastic differential equations driven by Wiener and Poisson processes. In order to do this, we focus on two model problems, the geometric Brownian motion and the linear stochastic oscillator, studied in the literature for stochastic differential equations only driven by a Wiener process. This essay covers theoretical as well as numerical investigations of jump - or more specifically, Poisson - processes and how they influence the above model problems.

  • 28.
    Bernland, Anders
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Acoustic shape optimization using cut finite elements2018In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 113, no 3, p. 432-449Article in journal (Refereed)
    Abstract [en]

    Fictitious domain methods are attractive for shape optimization applications, since they do not require deformed or regenerated meshes. A recently developed such method is the CutFEM approach, which allows crisp boundary representations and for which uniformly well-conditioned system matrices can be guaranteed. Here, we investigate the use of the CutFEM approach for acoustic shape optimization, using as test problem the design of an acoustic horn for favorable impedance-matching properties. The CutFEM approach is used to solve the Helmholtz equation, and the geometry of the horn is implicitly described by a level-set function. To promote smooth algorithmic updates of the geometry, we propose to use the nodal values of the Laplacian of the level-set function as design variables. This strategy also improves the algorithm's convergence rate, counteracts mesh dependence, and, in combination with Tikhonov regularization, controls small details in the optimized designs. An advantage with the proposed method is that the exact derivatives of the discrete objective function can be expressed as boundary integrals, as opposed to when using a traditional method that uses mesh deformations. The resulting horns possess excellent impedance-matching properties and exhibit surprising subwavelength structures, not previously seen, which are possible to capture due to the fixed mesh approach.

  • 29.
    Bernland, Anders
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    SHAPE OPTIMIZATION OF A COMPRESSION DRIVER PHASE PLUG2019In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 41, no 1, p. B181-B204Article in journal (Refereed)
    Abstract [en]

    A compression driver is an electro-acoustic transducer with considerably higher efficiency than direct radiating loudspeakers, thanks to the increased radiation resistance caused by a large vibrating diaphragm placed in a compression chamber with small openings. The transition section between compression chamber and output waveguide, the phase plug, must be carefully designed to avoid irregularities in the output sound pressure level (SPL) as a function of frequency. Here we present a shape optimization method based on an implicit level-set description and adjoint sensitivity analysis, which enables a large number of design parameters and vast design freedom. The CutFEM approach, a fictitious domain finite element method, removes the need for mesh updates and makes the method robust and computationally inexpensive. Numerical experiments for a generic annular diaphragm compression driver are presented, with optimized designs showing only minor frequency irregularities. Two different objective functions are considered: one for maximum SPL and one where the SPL is fitted to that of a hypothetical ideal design; the latter approach is found to be more effective in reducing irregularities. Visco-thermal boundary-layer losses are included in a post-processing step, and, though the influence of losses is clearly noticeable, the overall performance is similar and the optimized designs still outperform the original design.

  • 30.
    Billing, Erik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Servin, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Composer: A prototype multilingual model composition tool2013In: MODPROD2013: 7th MODPROD Workshop on Model-Based Product Development / [ed] Peter Fritzson, Umeå: Umeå universitet , 2013Conference paper (Other academic)
    Abstract [en]

    Composer - A prototype multilingual model composition tool

    Erik Billing and Martin Servin

    November 10, 2012

    Facing the task to design, simulate or optimize a complex system itis common to find models and data for the system expressed in differentformats, implemented in different simulation software tools. When a newmodel is developed, a target platform is chosen and existing componentsimplemented with different tools have to be converted. This results inunnecessary work duplication and lead times. The Modelica languageinitiative [2] partially solves this by allowing developers to move modelsbetween different tools following the Modelica standard. Another possi-bility is to exchange models using the Functional Mockup Interface (FMI)standard that allows computer models to be used as components in othersimulations, possibly implemented using other programming languages[1]. With the Modelica and FMI standards entering development, there isneed for an easy-to-use tool that supports design, editing and simulationof such multilingual systems, as well as for retracting system informationfor formulating and solving optimization problems.A prototype solution for a graphical block diagram tool for design, edit-ing, simulation and optimization of multilingual systems has been createdand evaluated for a specific system. The tool is named Composer [3].The block diagram representation should be generic, independent ofmodel implementations, have a standardized format and yet support effi-cient handling of complex data. It is natural to look for solutions amongmodern web technologies, specifically HTML5. The format for represent-ing two dimensional vector graphics in HTML5 is Scalable Vector Graphics(SVG). We combine the SVG format with the FMI standard. In a firststage, we take the XML-based model description of FMI as a form for de-scribing the interface for each component, in a language independent way.Simulation parameters can also be expressed on this form, and integratedas metadata into the SVG image. 

    The prototype, using SVG in conjunction with FMI, is implementedin JavaScript and allow creation and modification of block diagrams directly in the web browser. Generated SVG images are sent to the serverwhere they are translated to program code, allowing the simulation ofthe dynamical system to be executed using selected implementations. Analternative mode is to generate optimization problem from the systemdefinition and model parameters. The simulation/optimization result is 

    returned to the web browser where it is plotted or processed using otherstandard libraries.The fiber production process at SCA Packaging Obbola [4] is used asan example system and modeled using Composer. The system consists oftwo fiber production lines that produce fiber going to a storage tank [5].The paper machine is taking fiber from the tank as needed for production.A lot of power is required during fiber production and the purpose of themodel was to investigate weather electricity costs could be reduced byrescheduling fiber production over the day, in accordance with the electricity spot price. Components are implemented for dynamical simulationusing OpenModelica and for discrete event using Python. The Python implementation supports constraint propagation between components andoptimization over specified variables. Each component is interfaced as aFunctional Mock-up Unit (FMU), allowing components to be connectedand properties specified in language independent way. From the SVGcontaining the high-level system information, both Modelica and Pythoncode is generated and executed on the web server, potentially hosted ina high performance data center. More implementations could be addedwithout modifying the SVG system description.We have shown that it is possible to separate system descriptions onthe block diagram level from implementations and interface between thetwo levels using FMI. In a continuation of this project, we aim to integratethe FMI standard also for co-simulation, such that components implemented in different languages could be used together. One open questionis to what extent FMUs of the same component, but implemented withdifferent tools, will have the same model description. For the SVG-basedsystem description to be useful, the FMI model description must remainthe same, or at least contain a large overlap, for a single component implemented in different languages. This will be further investigated in futurework.

    References

    [1] Modelica Association. Functional mock-up interface, http://www.fmi-standard.org, November 2012.

    [2] Modelica Association. Modelica and the modelica association, http://www.modelica.org, November 2012.

    [3] Erik Billing and Martin Servin. Composer, http://imuit.cs.umu.se/composer, November 2012. [4] SCA Packaging. Sca packaging obbola, http://www.scapackaging.com, November 2012.

    [5] Patrik Törmänen and Hussein Jaffal. Reducing electricity cost - case study. Technical report, UMIT Research Lab, Umeå University, 2011.2

  • 31.
    Björklund, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    A Hierarchical POD Reduction Method of Finite Element Models with Application to Simulated Mechanical Systems2012Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    When simulating mechanical systems the flexibility of the components often has to be taken into account. This is particularly important for simulations when high detailed information is demanded, e.g. to calculate stresses. To this end the Finite Element Method (FEM) is often used. However the models can become very large, containing millions of degrees of freedom. Solving large linear systems are computationally demanding. Therefore ways of reducing the problem is often sought. These reduction does, however, remove much of the details that was to be investigated. In this thesis this problem is addressed by creating a reduction scheme, using Proper Orthogonal Decomposition (POD), that significantly reduces a problem but still captures much of the details. A novel method for enriching regular POD-based model reduction methods with hierarchically determined enrichment POD-modes is developed. The method is proposed and validated in a FEM application towards dynamical simulation. The enriched method is compared against a regular POD reduction technique. An numerical study is made of a model example of linear elasticity in a gearwheel. The numerical study suggests that the error of displacements is around ten times smaller, on average, when using the enriched basis compared to a reference basis of equal dimensionality consisting of only regular POD modes. Also it is shown that local quantities as the von Mises stress in a gearwheel tooth is preserved much better using the enriched basis. An a posteriori error estimate is proposed and proved for the static case, showing that the error is bound.

  • 32.
    Björnfot, Fredrik
    Umeå University, Faculty of Science and Technology, Department of Physics.
    GDP Growth Rate Nowcasting and Forecasting2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The main purpose of this project was to help Swedbank get a better understandingof how gross domestic product growth rate develops in the future froma data set of macroeconomic variables. Since GDP values are released long aftera quarter has ended Swedbank would like to have a model that could predictupcoming GDP from these data sets. This was solved by a combination ofgrowth rate predictions from a dynamic factor model, a vector autoregressivemodel and two machine learning models. The predictions were combined usinga weighting method called system averaging model where the model predictionwith least historical error receives the largest weight in the nal future prediction.In previous work a simple moving average model has been implementedto achieve this eect however there are several aws in a simple moving averagemodel. Most of these defects could in theory be avoided by using an exponentialweighting scheme instead. This resulted in the use of an exponentialweighting method that is used to calculate weights for future predictions. Themain conclusions from this project were that some predictions could get betterwhen removing bad performing models which had too large of a weight. Puttingtoo high weight on a single well performing model is also not optimal since thepredictions could get very unstable because of varying model performance. Theexponential weighting scheme worked well for some predictions however whenthe parameter , that controls how the weight is distributed between recent andhistorical errors, got too small a problem arose. Too few values were used toform the nal weights for the prediction and the estimate got unsteady results.

  • 33.
    Blanck, Andreas
    Umeå University, Faculty of Science and Technology, Department of Physics.
    American Option Price Approximation for Real-Time Clearing2018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    American-style options are contracts traded on financial markets. These are derivatives of some underlying security or securities that in contrast to European-style options allow their holders to exercise at any point before the contracts expire. However, this advantage aggravates the mathematical formulation of an option's value considerably, explaining why essentially no exact closed-formed pricing formulas exist. Numerous price approximation methods are although available, but their possible areas of application as well as performance, measured by speed and accuracy, differ. A clearing house offering real-time solutions are especially dependent on fast pricing methods to calculate portfolio risk, where accuracy is assumed to be an important factor to guarantee low-discrepancy estimations. Conversely, overly biased risk estimates may worsen a clearing house's ability to manage great losses, endangering the stability of a financial market it operates.

    The purpose of this project was to find methods with optimal performance and to investigate if price approximation errors induce biases in option portfolios' risk estimates. Regarding performance, a Quasi-Monte Carlo least squares method was found suitable for at least one type of exotic option. Yet none of the analyzed closed-form approximation methods could be assessed as optimal because of their varying strengths, where although the Binomial Tree model performed most consistently. Moreover, the answer to which method entails the best risk estimates remains inconclusive since only one set of parameters was used due to heavy calculations. A larger study involving a broader range of parameter values must therefore be performed in order to answer this reliably. However, it was revealed that large errors in risk estimates are avoided only if American standard options are priced with any of the analyzed methods and not when a faster European formula is employed. Furthermore, those that were analyzed can yield rather different risk estimates, implying that relatively large errors may arise if an inadequate method is applied.

  • 34. Boiveau, Thomas
    et al.
    Burman, Erik
    Claus, Susanne
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Fictitious domain method with boundary value correction using penalty-free Nitsche method2018In: Journal of Numerical Mathematics, ISSN 1570-2820, E-ISSN 1569-3953, Vol. 26, no 2, p. 77-95Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove optimal error estimates in the H1-norm and estimates suboptimal by 𝓞(h1/2) in the L2-norm. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical results are provided to corroborate the theoretical study.

  • 35.
    Brandl, Michael
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Newton’s Method for a Finite Element Approach to the Incompressible Navier-Stokes Equations2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The cG(1)cG(1)-method is a finite element method for solving the incompressible Navier-Stokes equations, using a splitting scheme and fixed-point iteration to resolve the nonlinear term u · ∇u. In this thesis, Newton’s method has been implemented on a formulation of the cG(1)cG(1)-method without splitting, resulting in equal results for the velocity and pressure computation, but higher computation times and slower convergence.

  • 36.
    Brändström, Samuel
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Invasion-analysis of stage-structured populations in temporally-varying environments2018Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Climate change may cause epidemic threats as species spreading human diseases invades previously unpopulated areas. A species can establish in new areas if they fulfills an invasion criterion. Two common invasion criteria are the long-term exponential growth-rate, r, and the basic reproduction number, R0, that measures the population's exponential growth in time and growth between generations respectively. Previous work have determined the long-term exponential growth-rate of the mosquito Aedes aegypti, a vector spreading dengue, zika and yellow fever, in Europe. However, in epidemiology r is rarely used as invasion criterion, which makes their results difficult to communicate and interpret. A more commonly used invasion criterion is the basic reproduction number R0. From this number, public health receives information about high-risk areas where they can vaccinate the population and prevent the mosquitoes from establishing by reducing breeding habitats. Here we extend the previous work by developing a method to calculate R0 for Aedes aegypti and verify it by the results from previous studies. Using R0 as invasion criterion we then predict the global distribution of Aedes aegypti during different climate change scenarios in the 21st century. One related to high emissions of greenhouse gases, RCP8.5, and one to low emissions, RCP2.6. We predict that the distribution of Aedes aegypti will expand towards higher latitudes at great speed during the 21st century assuming the high emission scenario RCP8.5. Assuming the low emission scenario RCP2.6, the distribution will not reach higher latitudes at the end of the 21st century. In Europe, the distribution covered 1.8 % in the beginning of the 20th century and at the end of the 21st century the distribution will cover 10 % assuming RCP8.5 and 2.0 % assuming RCP2.6. This work underscores the importance of reducing global warming and to take other preventive actions to avoid major epidemic outbreaks. Since we also provide instructions and software to calculate both r and R0 for stage-structured models in periodic environments, we anticipate that this work will support more studies of this kind to better understand the epidemic threats from climate change.

  • 37.
    Brännström, Åke
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Carlsson, Linus
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Int Inst Appl Syst Anal, Evolut & Ecol Program, A-2361 Laxenburg, Austria.
    Simpson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Int Inst Appl Syst Anal, Evolut & Ecol Program, A-2361 Laxenburg, Austria.
    On the convergence of the Escalator Boxcar Train2013In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 51, no 6, p. 3213-3231Article in journal (Refereed)
    Abstract [en]

    The Escalator Boxcar Train (EBT) is a numerical method that is widely used in theoretical biology to investigate the dynamics of physiologically structured population models, i.e., models in which individuals differ by size or other physiological characteristics. The method was developed more than two decades ago, but has so far resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate. In rigorously establishing the convergence of the EBT method, our results pave the way for wider acceptance of the EBT method beyond theoretical biology and constitutes an important step towards integration with established numerical schemes.

  • 38.
    Bujanović, Zvonimir
    et al.
    Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia.
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kressner, Daniel
    Institute of Mathematics, EPFL, Lausanne, Switzerland.
    A Householder-Based Algorithm for Hessenberg-Triangular Reduction2018In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 39, no 3, p. 1270-1294Article in journal (Refereed)
    Abstract [en]

    The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A nonvanishing fraction of the total flop-count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state-of-the-art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multithreaded BLAS. The design and evaluation of a parallel formulation is future work.

  • 39. Burman, Erik
    et al.
    Claus, Susanne
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, Andre
    Center for Biomedical Computing, Simula Research Laboratory, PO Box 134, NO-1325 Lysaker, Norway.
    CutFEM: Discretizing geometry and partial differential equations2015In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, p. 472-501Article in journal (Refereed)
    Abstract [en]

    We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer-aided design or image data from applied sciences. Both the treatment of boundaries and interfaces and the discretization of PDEs on surfaces are discussed and illustrated numerically.

  • 40.
    Burman, Erik
    et al.
    University College London.
    Claus, Susanne
    University College London.
    Massing, André
    Simula Res Lab, Fornebu, Norway.
    A Stabilized Cut Finite Element Method for the Three Field Stokes Problem2015In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 37, no 4, p. A1705-A1726Article in journal (Refereed)
    Abstract [en]

    We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure, and extra-stress tensor are discretized on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings numerically.

  • 41. Burman, Erik
    et al.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A cut finite element method for the Bernoulli free boundary value problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, p. 598-618Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H-1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H-1 norm. Finally, we present illustrating numerical results.

  • 42.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed)
    Abstract [en]

    We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represents parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.

  • 43.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats
    Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
    Larsson, Karl
    Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
    Shape optimization using the cut finite element method2018In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed)
    Abstract [en]

    We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re-meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two-dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.

  • 44. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A simple approach for finite element simulation of reinforced plates2018In: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 142, p. 51-60Article in journal (Refereed)
    Abstract [en]

    We present a new approach for adding Bernoulli beam reinforcements to Kirchhoff plates. The plate is discretised using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The beams are discretised by the CutFEM technique of letting the basis functions of the plate represent also the beams which are allowed to pass through the plate elements. This allows for a fast and easy way of assessing where the plate should be supported, for instance, in an optimization loop.

  • 45. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Augmented Lagrangian and Galerkin least-squares methods for membrane contact2018In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 114, no 11, p. 1179-1191Article in journal (Refereed)
    Abstract [en]

    In this paper, we propose a stabilized finite element method for the numerical solution of contact between a small deformation elastic membrane and a rigid obstacle. We limit ourselves to friction-free contact, but the formulation is readily extendable to more complex situations.

  • 46. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization2018In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 3, article id 035004Article in journal (Refereed)
    Abstract [en]

    Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection-diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson's equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.

  • 47. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The Penalty-Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem2017In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, no 6, p. 2523-2539Article in journal (Refereed)
    Abstract [en]

    We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 ( 2013), pp. 1295-1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.

  • 48.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, Andre
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Finite element approximation of the Laplace-Beltrami operator on a surface with boundary2019In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 141-172Article in journal (Refereed)
    Abstract [en]

    We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche's method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order k ≥ 1 in the energy and L2 norms that take the approximation of the surface and the boundary into account.

  • 49.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, Andre
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A cut discontinuous Galerkin method for the Laplace-Beltrami operator2017In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, no 1, p. 138-169Article in journal (Refereed)
    Abstract [en]

    We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in Rd . The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in Rd . The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then, a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighbouring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

  • 50. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zahedi, Sara
    Full gradient stabilized cut finite element methods for surface partial differential equations2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 278-296Article in journal (Refereed)
    Abstract [en]

    We propose and analyze a new stabilized cut finite element method for the Laplace Beltrami operator on a closed surface. The new stabilization term provides control of the full R-3 gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and L-2 norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

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