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  • 1.
    Amouzgar, Kaveh
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Metamodel based multi-objective optimization2015Licentiate thesis, comprehensive summary (Other academic)
    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.
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Cenanovic, Mirza
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Salomonsson, Kent
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Multi-objective optimization of material model parameters of an adhesive layer by using SPEA22015In: 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, p. 249-254Conference paper (Refereed)
    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.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Rashid, Asim
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Strömberg, Niclas
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Multi-Objective Optimization of a Disc Brake System by using SPEA2 and RBFN2013In: 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 , 2013Conference paper (Other academic)
    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.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Strömberg, N.
    Radial basis functions with a priori bias in comparisonwith a posteriori bias under multiple modeling criteriaIn: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488Article in journal (Other academic)
  • 5.
    Amouzgar, Kaveh
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Strömberg, Niclas
    University of West.
    An approach towards generating surrogate models by using RBFN with a apriori bias2014In: 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) , 2014Conference paper (Refereed)
    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.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. 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 bias2017In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, no 4, p. 1453-1469Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A finite element time relaxation method2011In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 349, no 5-6, p. 353-356Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A hierarchical NXFEM for fictitious domain simulations2011In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 86, no 4-5, p. 549-559Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity2009In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 198, no 41-44, p. 3352-3360Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A simple pressure stabilization method for the Stokes equation2008In: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 24, no 11, p. 1421-1430Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Discontinuous Galerkin methods for convection–diffusion problems with arbitrary Péclet number2000In: Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2003 / [ed] P. Neittaanmäki, T. Tiihonen, and P. Tarvainen, Berlin: Springer , 2000Conference paper (Refereed)
  • 12.
    Becker, Roland
    et al.
    Universität Heidelberg.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G
    Chalmers University of Technology.
    Energy norm a posteriori error estimation for discontinuous Galerkin methods2003In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 192, no 5-6, p. 723-733Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Stenberg, Rolf
    Helsinki University of Technology.
    A finite element method for domain decomposition with non-matching grids2003In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 37, no 2, p. 209-225Article in journal (Refereed)
    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
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umeå University, Sweden.
    Larsson, Karl
    Umeå University, Sweden.
    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 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
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    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 CutFEM2017In: Simulation for Additive Manufacturing 2017, Sinam 2017, International Center for Numerical Methods in Engineering (CIMNE), 2017, p. 208-209Conference paper (Refereed)
  • 16.
    Burman, Erik
    et al.
    University College London, United Kingdom.
    Elfverson, Daniel
    Umeå universitet, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats
    Umeå universitet, Sweden.
    Larsson, Karl
    Umeå universitet, Sweden.
    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.

  • 17. Burman, Erik
    et al.
    Fernandez, Miguel A.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Continuous interior penalty finite element method for Oseen's equations2006In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 44, no 3, p. 1248-1274Article in journal (Refereed)
    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.

  • 18.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A stabilized non-conforming finite element method for incompressible flow2006In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, no 23-24, p. 2881-2899Article in journal (Refereed)
    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.

  • 19.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    A unified stabilized method for Stokes' and Darcy's equations2007In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 198, no 1, p. 35-51Article in journal (Refereed)
    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.

  • 20.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom..
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Deriving Robust Unfitted Finite Element Methods from Augmented Lagrangian Formulations2017In: 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, p. 1-24Conference paper (Refereed)
    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.

  • 21.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems2004In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 193, no 15-16, p. 1437-1453Article in journal (Refereed)
    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.

  • 22.
    Burman, Erik
    et al.
    Ecole Polytechnique Fédérale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Edge stabilization for the generalized Stokes problem: A continuous interior penalty method2006In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 195, no 19-22, p. 2393-2410Article in journal (Refereed)
    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.

  • 23.
    Burman, Erik
    et al.
    University of Sussex.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method2010In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 199, no 41-44, p. 2680-2686Article in journal (Refereed)
    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.

  • 24.
    Burman, Erik
    et al.
    Department of Mathematics, University of Sussex.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method2012In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 62, no 4, p. 328-341Article in journal (Refereed)
    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. 

  • 25. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2011Report (Other academic)
    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.

  • 26.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2014In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 48, no 3, p. 859-874Article in journal (Refereed)
    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.

  • 27.
    Burman, Erik
    et al.
    Chalmers University of Technology and Göteborg University.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems2010In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 30, no 3, p. 870-885Article in journal (Refereed)
    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.

  • 28.
    Burman, Erik
    et al.
    Ecole Polytechnique Federale de Lausanne.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem2005In: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 21, no 5, p. 986-997Article in journal (Refereed)
    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.

  • 29.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Stabilized nonconforming finite element methods for data assimilation in incompressible flows2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 311, p. 1029-1050Article in journal (Refereed)
    Abstract [en]

    We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes equations. The method uses a primal dual structure that allows for the inclusion of nonstandard data. Error estimates are obtained that are optimal compared to the conditional stability of the ill-posed data assimilation problem.

  • 30. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    The edge stabilization method for finite elements in CFD2004In: Numerical mathematics and advanced applications / [ed] Feistauer, M; Dolejsi, V; Najzar, K, BERLIN: SPRINGER-VERLAG BERLIN , 2004, p. 196-203Conference paper (Other academic)
    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.

  • 31.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    A cut finite element method with boundary value correction2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 310, p. 633-657Article in journal (Refereed)
    Abstract [en]

    In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomée in [Math. Comp. 26 (1972), 869-879]. The cut finite element method is a fictitious domain method with Nitsche-type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

  • 32.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    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.

  • 33.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University.
    A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, p. 188-207Article in journal (Refereed)
    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. 

  • 34.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    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. 

  • 35.
    Burman, Erik
    et al.
    University College London, London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G
    Umeå University, Umeå, Sweden.
    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.

  • 36. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Institutionen för matematik och matematisk statistik, Umeå universitet.
    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.

  • 37.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Cut finite elements for convection in fractured domains2018In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain, which is a union of manifolds of different dimensions such that a d dimensional component always resides on the boundary of a d+1 dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

  • 38.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    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.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Finite element approximation of the Laplace–Beltrami operator on a surface with boundary2018In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245Article 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 (Formula presented.) in the energy and (Formula presented.) norms that take the approximation of the surface and the boundary into account. 

  • 39.
    Burman, Erik
    et al.
    University College London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umeå University, Sweden.
    Massing, André
    Umeå University, Sweden.
    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 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.

  • 40.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    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 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.

  • 41.
    Burman, Erik
    et al.
    Department of Mathematics, University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    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 problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 313, p. 362-374Article in journal (Refereed)
    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.

  • 42.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University.
    Zahedi, Sara
    KTH Royal Institute of Technology.
    Cut finite element methods for coupled bulk–surface problems2016In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, no 2, p. 203-231Article in journal (Refereed)
    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.

  • 43.
    Burman, Erik
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umea Universitet, Department of Mathematics and Mathematical Statistics, Umea, Sweden.
    Zahedi, Sara
    The Royal Institute of Technology (KTH), Department of Mathematics, Stockholm, Sweden.
    Stabilized CutFEM for the convection problem on surfaces2018In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, p. 1-37Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h3 / 2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3 / 4 order of convergence in L2. We also show that the condition number of the stiffness matrix is bounded by h- 2. Finally, our results are verified by numerical examples. 

  • 44.
    Cenanovic, Mirza
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Finite element methods for surface problems2017Doctoral thesis, comprehensive summary (Other academic)
    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.

  • 45.
    Cenanovic, Mirza
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Finite element methods on surfaces2015Licentiate thesis, comprehensive summary (Other academic)
    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.

  • 46.
    Cenanovic, Mirza
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Numerical error estimation for a TraceFEM membrane and distance function on P1 and P2 tetrahedraManuscript (preprint) (Other academic)
  • 47.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G,
    Umeå University.
    Minimal surface computation using a finite element method on an embedded surface2015In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, p. 502-512Article in journal (Refereed)
    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.

  • 48.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larsson, Mats G.
    Umeå University.
    Cut finite element modeling of linear membranes2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 98-111Article in journal (Refereed)
    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.

  • 49.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larsson, Mats G.
    Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinementManuscript (preprint) (Other academic)
    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.

  • 50. Chernov, Alexey
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    An hp-Nitsche’s Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes2011In: Spectral and High Order Methods for Partial Differential Equations / [ed] Jan S. Hesthaven and Einar M. Rønquist, 2011, p. 153-161Conference paper (Refereed)
    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.

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