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  • 1.
    Amouzgar, Kaveh
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Metamodel based multi-objective optimization2015Licentiatavhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    As a result of the increase in accessibility of computational resources and the increase in the power of the computers during the last two decades, designers are able to create computer models to simulate the behavior of a complex products. To address global competitiveness, companies are forced to optimize their designs and products. Optimizing the design needs several runs of computationally expensive simulation models. Therefore, using metamodels as an efficient and sufficiently accurate approximate of the simulation model is necessary. Radial basis functions (RBF) is one of the several metamodeling methods that can be found in the literature.

    The established approach is to add a bias to RBF in order to obtain a robust performance. The a posteriori bias is considered to be unknown at the beginning and it is defined by imposing extra orthogonality constraints. In this thesis, a new approach in constructing RBF with the bias to be set a priori by using the normal equation is proposed. The performance of the suggested approach is compared to the classic RBF with a posteriori bias. Another comprehensive comparison study by including several modeling criteria, such as problem dimension, sampling technique and size of samples is conducted. The studies demonstrate that the suggested approach with a priori bias is in general as good as the performance of RBF with a posteriori bias. Using the a priori RBF, it is clear that the global response is modeled with the bias and that the details are captured with radial basis functions.

    Multi-objective optimization and the approaches used in solving such problems are briefly described in this thesis. One of the methods that proved to be efficient in solving multi-objective optimization problems (MOOP) is the strength Pareto evolutionary algorithm (SPEA2). Multi-objective optimization of a disc brake system of a heavy truck by using SPEA2 and RBF with a priori bias is performed. As a result, the possibility to reduce the weight of the system without extensive compromise in other objectives is found.

    Multi-objective optimization of material model parameters of an adhesive layer with the aim of improving the results of a previous study is implemented. The result of the original study is improved and a clear insight into the nature of the problem is revealed.

  • 2.
    Amouzgar, Kaveh
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Cenanovic, Mirza
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Salomonsson, Kent
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Multi-objective optimization of material model parameters of an adhesive layer by using SPEA22015Inngår i: Advances in structural and multidisciplinary optimization: Proceedings of the 11th World Congress of Structural and Multidisciplinary Optimization (WCSMO-11) / [ed] Qing Li, Grant P Steven, Zhongpu (Leo) Zhang, The International Society for Structural and Multidisciplinary Optimization (ISSMO) , 2015, s. 249-254Konferansepaper (Fagfellevurdert)
    Abstract [en]

    The usage of multi material structures in industry, especially in the automotive industry are increasing. To overcome the difficulties in joining these structures, adhesives have several benefits over traditional joining methods. Therefore, accurate simulations of the entire process of fracture including the adhesive layer is crucial. In this paper, material parameters of a previously developed meso mechanical finite element (FE) model of a thin adhesive layer are optimized using the Strength Pareto Evolutionary Algorithm (SPEA2). Objective functions are defined as the error between experimental data and simulation data. The experimental data is provided by previously performed experiments where an adhesive layer was loaded in monotonically increasing peel and shear. Two objective functions are dependent on 9 model parameters (decision variables) in total and are evaluated by running two FEsimulations, one is loading the adhesive layer in peel and the other in shear. The original study converted the two objective functions into one function that resulted in one optimal solution. In this study, however, a Pareto frontis obtained by employing the SPEA2 algorithm. Thus, more insight into the material model, objective functions, optimal solutions and decision space is acquired using the Pareto front. We compare the results and show good agreement with the experimental data.

  • 3.
    Amouzgar, Kaveh
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Rashid, Asim
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Strömberg, Niclas
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Multi-Objective Optimization of a Disc Brake System by using SPEA2 and RBFN2013Inngår i: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 3B: 39th Design Automation ConferencePortland, Oregon, USA, August 4–7, 2013, New York: American Society of Mechanical Engineers , 2013Konferansepaper (Annet vitenskapelig)
    Abstract [en]

    Many engineering design optimization problems involve multiple conflicting objectives, which today often are obtained by computational expensive finite element simulations. Evolutionary multi-objective optimization (EMO) methods based on surrogate modeling is one approach of solving this class of problems. In this paper, multi-objective optimization of a disc brake system to a heavy truck by using EMO and radial basis function networks (RBFN) is presented. Three conflicting objectives are considered. These are: 1) minimizing the maximum temperature of the disc brake, 2) maximizing the brake energy of the system and 3) minimizing the mass of the back plate of the brake pad. An iterative Latin hypercube sampling method is used to construct the design of experiments (DoE) for the design variables. Next, thermo-mechanical finite element analysis of the disc brake, including frictional heating between the pad and the disc, is performed in order to determine the values of the first two objectives for the DoE. Surrogate models for the maximum temperature and the brake energy are created using RBFN with polynomial biases. Different radial basis functions are compared using statistical errors and cross validation errors (PRESS) to evaluate the accuracy of the surrogate models and to select the most accurate radial basis function. The multi-objective optimization problem is then solved by employing EMO using the strength Pareto evolutionary algorithm (SPEA2). Finally, the Pareto fronts generated by the proposed methodology are presented and discussed.

  • 4.
    Amouzgar, Kaveh
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Strömberg, N.
    Radial basis functions with a priori bias in comparisonwith a posteriori bias under multiple modeling criteriaInngår i: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488Artikkel i tidsskrift (Annet vitenskapelig)
  • 5.
    Amouzgar, Kaveh
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Strömberg, Niclas
    University of West.
    An approach towards generating surrogate models by using RBFN with a apriori bias2014Inngår i: Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2014 August 17-20, 2014, Buffalo, NY, USA, American Society of Mechanical Engineers (ASME) , 2014Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper, an approach to generate surrogate modelsconstructed by radial basis function networks (RBFN) with a prioribias is presented. RBFN as a weighted combination of radialbasis functions only, might become singular and no interpolationis found. The standard approach to avoid this is to add a polynomialbias, where the bias is defined by imposing orthogonalityconditions between the weights of the radial basis functionsand the polynomial basis functions. Here, in the proposed a prioriapproach, the regression coefficients of the polynomial biasare simply calculated by using the normal equation without anyneed of the extra orthogonality prerequisite. In addition to thesimplicity of this approach, the method has also proven to predictthe actual functions more accurately compared to the RBFNwith a posteriori bias. Several test functions, including Rosenbrock,Branin-Hoo, Goldstein-Price functions and two mathematicalfunctions (one large scale), are used to evaluate the performanceof the proposed method by conducting a comparisonstudy and error analysis between the RBFN with a priori and aposteriori known biases. Furthermore, the aforementioned approachesare applied to an engineering design problem, that ismodeling of the material properties of a three phase sphericalgraphite iron (SGI) . The corresponding surrogate models arepresented and compared

  • 6.
    Amouzgar, Kaveh
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering. School of Engineering Science, University of Skövde, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, School of Science and Technology, University of Örebro, Örebro, Sweden .
    Radial basis functions as surrogate models with a priori bias in comparison with a posteriori bias2017Inngår i: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, nr 4, s. 1453-1469Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In order to obtain a robust performance, the established approach when using radial basis function networks (RBF) as metamodels is to add a posteriori bias which is defined by extra orthogonality constraints. We mean that this is not needed, instead the bias can simply be set a priori by using the normal equation, i.e. the bias becomes the corresponding regression model. In this paper we demonstrate that the performance of our suggested approach with a priori bias is in general as good as, or even for many test examples better than, the performance of RBF with a posteriori bias. Using our approach, it is clear that the global response is modelled with the bias and that the details are captured with radial basis functions. The accuracy of the two approaches are investigated by using multiple test functions with different degrees of dimensionality. Furthermore, several modeling criteria, such as the type of radial basis functions used in the RBFs, dimension of the test functions, sampling techniques and size of samples, are considered to study their affect on the performance of the approaches. The power of RBF with a priori bias for surrogate based design optimization is also demonstrated by solving an established engineering benchmark of a welded beam and another benchmark for different sampling sets generated by successive screening, random, Latin hypercube and Hammersley sampling, respectively. The results obtained by evaluation of the performance metrics, the modeling criteria and the presented optimal solutions, demonstrate promising potentials of our RBF with a priori bias, in addition to the simplicity and straight-forward use of the approach.

  • 7.
    Becker, Roland
    et al.
    Université de Pau.
    Burman, Erik
    University of Sussex.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A finite element time relaxation method2011Inngår i: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 349, nr 5-6, s. 353-356Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We discuss a finite element time-relaxation method for high Reynolds number flows. The method uses local projections on polynomials defined on macroelements of each pair of two elements sharing a face. We prove that this method shares the optimal stability and convergence properties of the continuous interior penalty (CIP) method. We give the formulation both for the scalar convection-diffusion equation and the time-dependent incompressible Euler equations and the associated convergence results. This note finishes with some numerical illustrations.

  • 8.
    Becker, Roland
    et al.
    Université de Pau et des Pays de l'Adour.
    Burman, Erik
    University of Sussex Falmer.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A hierarchical NXFEM for fictitious domain simulations2011Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 86, nr 4-5, s. 549-559Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We suggest a fictitious domain method, based on the Nitsche XFEM method of (Comput. Meth. Appl. Mech. Engrg 2002; 191: 5537-5552), that employs a band of elements adjacent to the boundary. In contrast, the classical fictitious domain method uses Lagrange multipliers on a line (surface) where the boundary condition is to be enforced. The idea can be seen as an extension of the Chimera method of (ESAIM: Math. Model Numer. Anal. 2003; 37: 495-514), but with a hierarchical representation of the discontinuous solution field. The hierarchical formulation is better suited for moving fictitious boundaries since the stiffness matrix of the underlying structured mesh can be retained during the computations. Our technique allows for optimal convergence properties irrespective of the order of the underlying finite element method.

  • 9.
    Becker, Roland
    et al.
    Université de Pau et des Pays de l’Adour.
    Burman, Erik
    University of Sussex.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity2009Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 198, nr 41-44, s. 3352-3360Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this note we propose a finite element method for incompressible (or compressible) elasticity problems with discontinuous modulus of elasticity (or, if compressible, Poisson's ratio). The problem is written on mixed form using P(1)-continuous displacements and elementwise P(0) pressures, leading to the possibility of eliminating the pressure beforehand in the compressible case. In the incompressible case, the method is augmented by a stabilization term, penalizing the pressure jumps. We show a priori error estimates under certain regularity hypothesis. In particular we prove that if the exact solution is sufficiently smooth in each subdomain then the convergence order is optimal.

  • 10.
    Becker, Roland
    et al.
    Université de Pau et des Pays de l'Adour.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A simple pressure stabilization method for the Stokes equation2008Inngår i: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 24, nr 11, s. 1421-1430Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we consider a stabilization method for the Stokes problem, using equal-order interpolation of the pressure and velocity, which avoids the use of the mesh size parameter in the stabilization term. We show that our approach is stable for equal-order interpolation in the case of piecewise linear and piecewise quadratic polynomials on triangles. In the case of linear polynomials, we retrieve a well-known idea of using mass lumping as a stabilization mechanism.

  • 11. Becker, Roland
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Discontinuous Galerkin methods for convection–diffusion problems with arbitrary Péclet number2000Inngår i: Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2003 / [ed] P. Neittaanmäki, T. Tiihonen, and P. Tarvainen, Berlin: Springer , 2000Konferansepaper (Fagfellevurdert)
  • 12.
    Becker, Roland
    et al.
    Universität Heidelberg.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G
    Chalmers University of Technology.
    Energy norm a posteriori error estimation for discontinuous Galerkin methods2003Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 192, nr 5-6, s. 723-733Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we present a residual-based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous Galerkin approximations of elliptic problems. The theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.

  • 13.
    Becker, Roland
    et al.
    University of Heidelberg.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Stenberg, Rolf
    Helsinki University of Technology.
    A finite element method for domain decomposition with non-matching grids2003Inngår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 37, nr 2, s. 209-225Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this note, we propose and analyse a method for handling interfaces between nonmatching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

  • 14.
    Burman, Erik
    et al.
    University College London, Gower Street, UK.
    Elfverson, Daniel
    Umeå University, Sweden.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G.
    Umeå University, Sweden.
    Larsson, Karl
    Umeå University, Sweden.
    A cut finite element method for the Bernoulli free boundary value problem2017Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, s. 598-618Artikkel i tidsskrift (Fagfellevurdert)
    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 H1 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 H1norm. Finally, we present illustrating numerical results.

  • 15.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, Gower Street, London, United Kingdom.
    Elfverson, Daniel
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Shape and topology optimization using CutFEM2017Inngår i: Simulation for Additive Manufacturing 2017, Sinam 2017, International Center for Numerical Methods in Engineering (CIMNE), 2017, s. 208-209Konferansepaper (Fagfellevurdert)
    Abstract [en]

    We present a shape and topology optimization method based on the cut finite element method, see [1],[2], and [3], for the optimal compliance problem in linear elasticity and problems involving restrictionson the stresses.The elastic domain is defined by a level-set function, and the evolution of the domain is obtained bymoving the level-set along a velocity field using a transport equation. The velocity field is defined tobe the largest decreasing direction of the shape derivative that resides in a certain Hilbert space and iscomputed by solving an elliptic problem, associated with the bilinear form in the Hilbert space, with theshape derivative as right hand side. The velocity field may thus be viewed as the Riesz representationof the shape derivative on the chosen Hilbert space.We thus obtain a coupled problem involving three partial differential equations: (1) the elasticity problem,(2) the elliptic problem that determines the velocity field, and (3) the transport problem for thelevelset function. The elasticity problem is solved using a cut finite element method on a fixed backgroundmesh, which completely avoids re–meshing when the domain is updated. The levelset functionand the velocity field is approximated by standard conforming elements on the background mesh. Wealso employ higher order cut approximations including isogeometric analysis for the elasticity problem.In this case the levelset function and the velocity field are represented using linear elements on a refinedmesh in order to simplify the geometric and quadrature computations on the cut elements. To obtain astable method, stabilization terms are added in the vicinity of the cut elements at the boundary, whichprovides control of the variation of the solution in the vicinity of the boundary. We present numericalexamples illustrating the performance of the method.We also study an anisotropic material model that accounts for the orientation of the layers in an additivemanufacturing process and by including the orientation in the optimization problem we determine theoptimal choice of orientation.We present numerical results including test problems and engineering applications in additive manufacturing.

    References

    [1] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: discretizing geometryand partial differential equations. Internat. J. Numer. Methods Engrg., 104(7):472–501, 2015.

    [2] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. Shape optimization using thecut finite element method. Technical report, 2016. arXiv:1611.05673.

    [3] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. A cut finite element method forthe Bernoulli free boundary value problem. Comput. Methods Appl. Mech. Engrg., 317:598–618,2017.

  • 16. Burman, Erik
    et al.
    Fernandez, Miguel A.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Continuous interior penalty finite element method for Oseen's equations2006Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 44, nr 3, s. 1248-1274Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [ Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207 - 216] to Oseen's equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/ velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L-2-control of the jump of the gradient over element faces ( edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.

  • 17.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A stabilized non-conforming finite element method for incompressible flow2006Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, nr 23-24, s. 2881-2899Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we extend the recently introduced edge stabilization method to the case of non-conforming finite element approximations of the linearized Navier-Stokes equation. To get stability also in the convective dominated regime we add a term giving L-2-control of the jump in the gradient over element boundaries. An a priori error estimate that is uniform in the Reynolds number is proved and some numerical examples are presented.

  • 18.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A unified stabilized method for Stokes' and Darcy's equations2007Inngår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 198, nr 1, s. 35-51Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We use the lowest possible approximation order, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Stokes equation and Darcy's equation, applying an edge stabilization term to avoid locking. We prove that the formulation satisfies the discrete inf-sup condition, we prove an optimal a priori error estimate for both problems. The formulation is then extended to the coupled case using a Nitsche-type weak formulation allowing for different meshes in the two subdomains. Finally, we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach.

  • 19.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom..
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Deriving Robust Unfitted Finite Element Methods from Augmented Lagrangian Formulations2017Inngår i: Geometrically Unfitted Finite Element Methods and Applications / [ed] Bordas, Stéphane P. A.; Burman, Erik; Larson, Mats G.; Olshanskii, Maxim A., Cham: Springer International Publishing , 2017, s. 1-24Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper we will discuss different coupling methods suitable for use in the framework of the recently introduced CutFEM paradigm, cf. Burman et al. (Int. J. Numer. Methods Eng. 104(7):472–501, 2015). In particular we will consider mortaring using Lagrange multipliers on the one hand and Nitsche’s method on the other. For simplicity we will first discuss these methods in the setting of uncut meshes, and end with some comments on the extension to CutFEM. We will, for comparison, discuss some different types of problems such as high contrast problems and problems with stiff coupling or adhesive contact. We will review some of the existing methods for these problems and propose some alternative methods resulting from crossovers from the Lagrange multiplier framework to Nitsche’s method and vice versa.

  • 20.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems2004Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 193, nr 15-16, s. 1437-1453Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we recall a stabilization technique for finite element methods for convection-diffusion-reaction equations, originally proposed by Douglas and Dupont [Computing Methods in Applied Sciences, Springer-Verlag, Berlin, 1976]. The method uses least square stabilization of the gradient jumps a across element boundaries. We prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates. We address the question of monotonicity of discrete Solutions and present some numerical examples illustrating the theoretical results.

  • 21.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Edge stabilization for the generalized Stokes problem: A continuous interior penalty method2006Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, nr 19-22, s. 2393-2410Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this note we introduce and analyze a stabilized finite element method for the generalized Stokes equation. Stability is obtained by adding a least squares penalization of the gradient jumps across element boundaries. The method can be seen as a higher order version of the Brezzi-Pitkdranta penalty stabilization [F. Brezzi, J. Pitkaranta, On the stabilization of finite element approximations of the Stokes equations, in: W. Hackbusch (Ed.), Efficient Solution of Elliptic Systems, Vieweg, 1984], but gives better resolution on the boundary for the Stokes equation than does classical Galerkin least-squares formulation. We prove optimal and quasi-optimal convergence properties for Stokes' problem and for the porous media models of Darcy and Brinkman. Some numerical examples are given.

  • 22.
    Burman, Erik
    et al.
    University of Sussex.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method2010Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 199, nr 41-44, s. 2680-2686Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We propose a fictitious domain method where the mesh is cut by the boundary. The primal solution is computed only up to the boundary; the solution itself is defined also by nodes outside the domain, but the weak finite element form only involves those parts of the elements that are located inside the domain. The multipliers are defined as being element-wise constant on the whole (including the extension) of the cut elements in the mesh defining the primal variable. Inf-sup stability is obtained by penalizing the jump of the multiplier over element faces. We consider the case of a polygonal domain with possibly curved boundaries. The method has optimal convergence properties.

  • 23.
    Burman, Erik
    et al.
    Department of Mathematics, University of Sussex.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method2012Inngår i: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, nr 4, s. 328-341Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We extend the classical Nitsche type weak boundary conditions to a fictitious domain setting. An additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh. Optimal a priori error estimates in the H1- and L2-norms are proved as well as an upper bound on the condition number of the system matrix. 

  • 24. Burman, Erik
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Maskinteknik. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2011Rapport (Annet vitenskapelig)
    Abstract [en]

    We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf–sup stable and stabilized velocity pressure pairs are considered.

  • 25.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Maskinteknik. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2014Inngår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 48, nr 3, s. 859-874Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We extend the classical Nitsche type weak boundary conditions to a fictitious domain setting. An additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh. Optimal a priori error estimates in the H 1- and L 2-norms are proved as well as an upper bound on the condition number of the system matrix.

  • 26.
    Burman, Erik
    et al.
    Chalmers University of Technology and Göteborg University.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems2010Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 30, nr 3, s. 870-885Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we propose a class of jump-stabilized Lagrange multiplier methods for the finite-element solution of multidomain elliptic partial differential equations using piecewise-constant or continuous piecewise-linear approximations of the multipliers. For the purpose of stabilization we use the jumps in derivatives of the multipliers or, for piecewise constants, the jump in the multipliers themselves, across element borders. The ideas are illustrated using Poisson's equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

  • 27.
    Burman, Erik
    et al.
    Ecole Polytechnique Federale de Lausanne.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem2005Inngår i: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 21, nr 5, s. 986-997Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We stabilize the nonconforming Crouzeix-Raviart element for the Darcy-Stokes problem with terms motivated by a discontinuous Galerkin approach. Convergence of the method is shown, also in the limit of vanishing viscosity. Finally, some numerical examples verifying the theoretical predictions are presented.

  • 28. Burman, Erik
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    The edge stabilization method for finite elements in CFD2004Inngår i: Numerical mathematics and advanced applications / [ed] Feistauer, M; Dolejsi, V; Najzar, K, BERLIN: SPRINGER-VERLAG BERLIN , 2004, s. 196-203Konferansepaper (Annet vitenskapelig)
    Abstract [en]

    We give a brief overview of our recent work on the edge stabilization method for flow problems. The application examples are convection-diffusion, with small diffusion parameter, and a generalized Stokes model.

  • 29.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering. Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling.
    Larson, Mats G.
    Umeå University.
    A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator2015Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, s. 188-207Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider solving the Laplace–Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.

    The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method. 

  • 30. Burman, Erik
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G.
    Institutionen för matematik och matematisk statistik, Umeå universitet.
    The penalty-free Nitsche Method and nonconforming finite elements for the Signorini problem2017Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, nr 6, s. 2523-2539Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 31.
    Burman, Erik
    et al.
    University College London, UK.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G.
    Umeå University, Sweden.
    Massing, André
    Umeå University, Sweden.
    A cut discontinuous Galerkin method for the Laplace–Beltrami operator2017Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, nr 1, s. 138-169Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in R. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in R. 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.

  • 32.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Zahedi, Sara
    Department of Mathematics, KTH, Stockholm, Sweden.
    Full gradient stabilized cut finite element methods for surface partial differential equations2016Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, s. 278-296Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 33.
    Burman, Erik
    et al.
    Department of Mathematics, University College London.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University.
    Stenberg, Rolf
    Department of Mathematics and Systems Analysis, Aalto University.
    Galerkin least squares finite element method for the obstacle problem2017Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 313, s. 362-374Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We construct a consistent multiplier free method for the finite element solution of the obstacle problem. The method is based on an augmented Lagrangian formulation in which we eliminate the multiplier by use of its definition in a discrete setting. We prove existence and uniqueness of discrete solutions and optimal order a priori error estimates for smooth exact solutions. Using a saturation assumption we also prove an a posteriori error estimate. Numerical examples show the performance of the method and of an adaptive algorithm for the control of the discretization error.

  • 34.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering. Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling.
    Larson, Mats G.
    Umeå University.
    Zahedi, Sara
    KTH Royal Institute of Technology.
    Cut finite element methods for coupled bulk–surface problems2016Inngår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, nr 2, s. 203-231Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and L2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

  • 35.
    Cenanovic, Mirza
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Finite element methods for surface problems2017Doktoravhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    The purpose of this thesis is to further develop numerical methods for solving surface problems by utilizing tangential calculus and the trace finite element method. Direct computation on the surface is possible by the use of tangential calculus, in contrast to the classical approach of mapping 2D parametric surfaces to 3D surfaces by means of differential geometry operators. Using tangential calculus, the problem formulation is only dependent on the position and normal vectors of the 3D surface. Tangential calculus thus enables a clean, simple and inexpensive formulation and implementation of finite element methods for surface problems. Meshing techniques are greatly simplified from the end-user perspective by utilizing an unfitted finite element method called the Trace Finite Element Method, in which the basic idea is to embed the surface in a higher dimensional mesh and use the shape functions of this background mesh for the discretization of the partial differential equation. This method makes it possible to model surfaces implicitly and solve surface problems without the need for expensive meshing/re-meshing techniques especially for moving surfaces or surfaces embedded in 3D solids, so called embedded interface problems. Using these two approaches, numerical methods for solving three surface problems are proposed: 1) minimal surface problems, in which the form that minimizes the mean curvature was computed by iterative update of a level-set function discretized using TraceFEM and driven by advection, for which the velocity field was given by the mean curvature flow, 2) elastic membrane problems discretized using linear and higher order TraceFEM, which makes it straightforward to embed complex geometries of membrane models into an elastic bulk for reinforcement and 3) stabilized, accurate vertex normal and mean curvature estimation with local refinement on triangulated surfaces. In this thesis the basics of the two main approaches are presented, some aspects such as stabilization and surface reconstruction are further developed, evaluated and numerically analyzed, details on implementations are provided and the current state of work is presented.

  • 36.
    Cenanovic, Mirza
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Finite element methods on surfaces2015Licentiatavhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    The purpose of this thesis is to improve numerical simulations of surface problems. Two novel computational concepts are analyzed and applied on two surface problems; minimal surface problems and elastic membrane problems. The concept of tangential projection implies that direct computation on the surface is made possible compared to the classical approach of mapping 2D parametric surfaces to 3D surfaces by means of differential geometry operators. The second concept presented is the cut finite element method, in which the basic idea of discretization is to embed the d- 1-dimensional surface in a d-dimensional mesh and use the basis functions of a higher dimensional mesh but integrate over the surface. The aim of this thesis is to present the basics of the two main approaches and to provide details on the implementation.

  • 37.
    Cenanovic, Mirza
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Numerical error estimation for a TraceFEM membrane and distance function on P1 and P2 tetrahedraManuskript (preprint) (Annet vitenskapelig)
  • 38.
    Cenanovic, Mirza
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering. Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Maskinteknik. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G,
    Umeå University.
    Minimal surface computation using a finite element method on an embedded surface2015Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, nr 7, s. 502-512Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We suggest a finite element method for finding minimal surfaces based on computing a discrete Laplace–Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is carried out on the piecewise planar isosurface using the shape functions from the background three-dimensional mesh used to represent the distance function. A recently suggested stabilized scheme for finite element approximation of the mean curvature vector is a crucial component of the method.

  • 39.
    Cenanovic, Mirza
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larsson, Mats G.
    Umeå University.
    Cut finite element modeling of linear membranes2016Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, s. 98-111Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus, i.e., with the equilibrium equations pointwise projected onto the tangent plane of the surface to create a pointwise planar problem in the tangential direction. Both free membranes and membranes coupled to 3D elasticity are considered. The discretization of the membrane comes from a Galerkin method using the restriction of 3D basis functions (linear or trilinear) to the surface representing the membrane. In the case of coupling to 3D elasticity, we view the membrane as giving additional stiffness contributions to the standard stiffness matrix resulting from the discretization of the three-dimensional continuum.

  • 40.
    Cenanovic, Mirza
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larsson, Mats G.
    Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinementManuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L2-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L2-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare or results to previously proposed approaches.

  • 41. Chernov, Alexey
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    An hp-Nitsche’s Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes2011Inngår i: Spectral and High Order Methods for Partial Differential Equations / [ed] Jan S. Hesthaven and Einar M. Rønquist, 2011, s. 153-161Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper we propose an hp-Nitsche’s method for the finite element solution of interface elliptic problems using non-matched unstructured meshes of triangles and parallelograms in  2 and tetrahedra and parallelepipeds in  3. We obtain an explicit lower bound for the penalty weighting function in terms of the local inverse inequality constant. We prove a priori error estimates which are explicit in the mesh size h and in the polynomial degree p. The error bound is optimal in h and suboptimal in polynomial degree by p 1∕2.

  • 42. Eriksson, Kenneth
    et al.
    Estep, Don
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Johnson, Claes
    Introduction to Adaptive Methods for Differential Equations1995Inngår i: Acta Numerica, ISSN 0962-4929, E-ISSN 1474-0508, Vol. 4, s. 105-158Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).

    When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

  • 43. Eriksson, Kenneth
    et al.
    Estep, Don
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Johnson, Claes
    Introduction to computational methods for differential equations1995Inngår i: Theory and numerics of ordinary and partial differential equations / [ed] M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, Oxford: Oxford University Press , 1995, s. 77-122Kapittel i bok, del av antologi (Fagfellevurdert)
  • 44. Eriksson, Kenneth
    et al.
    Estep, Donald
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Johnson, Claes
    Computational differential equations1996Bok (Fagfellevurdert)
  • 45.
    Gimbel, F.
    et al.
    University Siegen.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Suttmeier, F. T.
    University Siegen.
    An adaptive low-order FE-scheme for Stokes flow with cavitation2010Inngår i: Journal of Numerical Mathematics, ISSN 1570-2820, E-ISSN 1569-3953, Vol. 18, nr 3, s. 177-185Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this note we derive a posteriori error bounds for FE-discretisations for a fluid problem with cavitation. The underlying model is the Stokes system together with an inequality constraint for the pressure. In order to avoid suboptimal behavior of the error bounds we propose to employ a Lagrange setting yielding an improved estimate. Numerical tests confirm our theoretical results.

  • 46.
    Gustafsson, Tim
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Jansson, Johan
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Development and Implementation of an Anisotropic Material Model based on the Fibre Orientation in GFRPs2016Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
    Abstract [en]

    Based on the fibre orientation distribution of a glass fibre reinforced polymer (GFRP) an anisotropic material model is developed and implemented in the commercial Finite Element code LS-Dyna. The material model is coupled with the fibre orientation extracted from flow filling simulations performed in Moldex3D. Central mathematical models used are the Eshelby-Mori-Tanaka model for calculating a composites effective stiffness, Advani and Tucker’s orientation averaging method for considering the fibre orientation tensors, a Lagrange polynomial controlled radial return algorithm for the application of nonlinear hardening and finally Hashin’s damage initiation criterion to establish the initiation and evolution of damage. The material model is calibrated by adjusting Lagrange polynomial data points, until the simulation response corresponds to that of experimental tensile tests. Two different glass fibre reinforced composites are evaluated, and it is concluded that the model can capture the behaviour of the experimental curves with high accuracy. By validation against experimental results, it is shown that the material model performs well. The model is also compared to other advanced commercial material modelling software’s.

  • 47.
    Hansbo, Anita
    et al.
    Department of Informatics and Mathematics, University of Trollhättan Uddevalla.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A finite element method for the simulation of strong and weak discontinuities in solid mechanics2004Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 193, nr 33-35, s. 3523-3540Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we introduce and analyze a finite element method for elasticity problems with interfaces. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code. For the case of linear elasticity, we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. We present numerical examples for the linear case as well as for contact and crack propagation model problems.

  • 48. Hansbo, Anita
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    An unfitted finite element method, based on Nitsche's method, for elliptic interface problems2002Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 191, nr 47-48, s. 5537-5552Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we propose a method for the finite element solution of elliptic interface problem, using an approach due to Nitsche. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. Further, we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples.

  • 49. Hansbo, Anita
    et al.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Larson, Mats G
    A finite element method on composite grids based on Nitsche's method2003Inngår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 37, nr 3, s. 495-514Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we propose a finite element method for the approximation of second order elliptic problems on composite grids. The method is based on continuous piecewise polynomial approximation on each grid and weak enforcement of the proper continuity at an artificial interface defined by edges (or faces) of one the grids. We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions, and present some numerical examples.

  • 50.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    A Crank-Nicolson type space-time finite element method for computing on moving meshes2000Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 159, nr 2, s. 274-289Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, a space-time finite element method for evolution problems that is second-order accurate in both space and time is proposed. For convection dominated problems, the elements may be aligned along the characteristics in space-time, which results in a Crank-Nicolson method along the characteristics. The method is also suitable as an alternative to other moving mesh methods for problems in deforming domains. Numerical examples dealing with diffusion and convection are given.

1234 1 - 50 of 159
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