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  • 1.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    A modified Schwarz-Christoffel mapping for regions with piecewise smooth boundaries2008In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 213, no 1, p. 56-70Article in journal (Refereed)
    Abstract [en]

    A method where polygon corners in Schwarz-Christoffel mappings are rounded, is used to construct mappings from the upper half-plane to regions bounded by arbitrary piecewise smooth curves. From a given curve, a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points. The Schwarz-Christoffel mapping for that polygon is then constructed and modified to round the corners.Since such a modification causes effects on the polygon outside the rounded corners, the parameters in the mapping have to be re-determined. This is done by comparing side-lengths in tangent polygons to the given curve and the curve produced by the modified Schwarz-Christoffel mapping. The set of equations that this comparison gives, can normally be solved using a quasi--Newton method.The resulting function maps the upper half--plane on a region bounded by a curve that apart from possible vertices is smooth, i.e., one time continuously differentiable, that passes through the tangent points on the given curve, has the same direction as the given curve in these points and changes direction monotonically between them. Furthermore, where the original curve has a vertex, the constructed curve has a vertex with the same inner angle.The method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines, such as channels with parallel walls at the ends. These properties are kept in the region produced by the constructed mapping.

  • 2.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    Modified Schwarz-Christoffel mappings using approximate curve factors2009In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 233, no 4, p. 1117-1127Article in journal (Refereed)
    Abstract [en]

    The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.

    In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C1 regularity, C regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.

    Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C) boundary curve can be achieved.

  • 3.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    Numerical Conformal Mappings for Regions Bounded by Smooth Curves2006Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In many applications, conformal mappings are used to transform twodimensional regions into simpler ones. One such region for which conformal mappings are needed is a channel bounded by continuously differentiable curves. In the applications that have motivated this work, it is important that the region an approximate conformal mapping produces, has this property, but also that the direction of the curve can be controlled, especially in the ends of the channel.

    This thesis treats three different methods for numerically constructing conformal mappings between the upper half–plane or unit circle and a region bounded by a continuously differentiable curve, where the direction of the curve in a number of control points is controlled, exact or approximately.

    The first method is built on an idea by Peter Henrici, where a modified Schwarz–Christoffel mapping maps the upper half–plane conformally on a polygon with rounded corners. His idea is used in an algorithm by which mappings for arbitrary regions, bounded by smooth curves are constructed.

    The second method uses the fact that a Schwarz–Christoffel mapping from the upper half–plane or unit circle to a polygon maps a region Q inside the half–plane or circle, for example a circle with radius less than 1 or a sector in the half–plane, on a region Ω inside the polygon bounded by a smooth curve. Given such a region Ω, we develop methods to find a suitable outer polygon and corresponding Schwarz–Christoffel mapping that gives a mapping from Q to Ω.

    Both these methods use the concept of tangent polygons to numerically determine the coefficients in the mappings.

    Finally, we use one of Don Marshall’s zipper algorithms to construct conformal mappings from the upper half–plane to channels bounded by arbitrary smooth curves, with the additional property that they are parallel straight lines when approaching infinity.

  • 4.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    Numerical conformal mappings for waveguides2009Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Acoustic or electro-magnetic scattering in a waveguide with varying direction and cross-section can be re-formulated as a two-dimensional scattering problem, provided that the variations take place in only one dimension at a time. By using the so-called Building Block Method, it is possible to construct the scattering properties of a combination of scatterers when the properties of each scatterer are known. Hence, variations in the waveguide geometry or in the boundary conditions can be treated one at a time.

    Using the Building Block Method, the problem takes the form of the Helmholtz equation for stationary waves in a waveguide of infinite length and with smoothly varying geometry and boundary conditions. A conformal mapping is used to transform the problem into a corresponding problem in a straight horizontal waveguide, and by expanding the field in Fourier trigonometric series, the problem can be reformulated as an infinite-dimensional ordinary differential equation. From this, numerically solvable differential equations for the reflection and transmission operators are derived.

    To be applicable in the Building Block Method, the numerical conformal mapping must be constructed such that the direction of the boundary curve can be controlled. At the channel ends, it is an indispensable requirement, that the two boundary curves are (at least) asymptotically parallel and straight. Furthermore, to achieve bounded operators in the differential equations, the boundary curves must satisfy different regularity conditions, depending on the boundary conditions.

    In this work, several methods to accomplish such conformal mappings are presented. The Schwarz–Christoffel mapping, which is a natural starting point and for which also efficient numerical software exists, can be modified in different ways in order to achieve polygons with rounded corners. We present algorithms by which the parameters in the mappings can be determined after such modifications. We show also how the unmodified Schwarz–Christoffel mapping can be used for regions with a smooth boundary. This is done by constructing an appropriate outer polygon to the considered region.

    Finally, we introduce one method that is not Schwarz–Christoffel-related, by showing how one of the so-called zipper algorithms can be used for waveguides.

  • 5.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    Numerical Conformal Mappings for Waveguides2010In: Computational Mathematics: Theory, Methods and Applications, Hauppauge NY, USA: Nova Science Publishers , 2010Chapter in book (Other (popular science, discussion, etc.))
    Abstract [en]

    Acoustic or electro-magnetic scattering in a waveguide with  varying direction and cross-section can, if the variations takes  place in only one dimension at a time be re-formulated as a  two-dimensional scattering problem. By using the so-called  Building Block Method, it is possible to construct the  scattering properties of a combination of scatterers when the  properties of each scatterer are known. Hence, variations in the  waveguide geometry or in the boundary conditions can be treated   one at a time.  We consider in this work acoustic scattering, but the same  techniques can be used for both electro-magnetic and some quantum  scattering problems.  By suppressing the time dependence and by using the Building Block  Method, the problem takes the form of the Helmholtz equation in a  waveguide of infinite length and with smoothly varying geometry and  boundary conditions.  A conformal mapping is used to transform the  problem into a corresponding problem in a straight horizontal  channel, and by expanding the field in Fourier trigonometric series,  the problem can be reformulated as an infinite-dimensional ordinary  differential equation. From this, numerically solvable differential  equations for the reflection and transmission operators are  derived.  To be applicable in the Building Block Method, the numerical  conformal mapping must be constructed such that the direction of the  boundary curve can be controlled. At the channel ends, it is an  indispensable requirement, that the two boundary curves are (at least)  asymptotically parallel and straight. Furthermore, to achieve  bounded operators in the differential equations, the boundary curves  must satisfy different regularity conditions, depending on the  properties of the boundary.  Several methods to accomplish such conformal mappings are  presented. The Schwarz-Christoffel mapping, which is a natural starting point and for which  also efficient numerical software exists, can be modified in  different ways to round the polygon corners, and we show algorithms  by which the parameter problem can be solved after such  modifications. It is also possible to use the unmodified Schwarz-Christoffel mapping for  regions with smooth boundary, by constructing an appropriate outer  polygon to the considered region.  Finally, we show how a so-called  zipper algorithm can be used for waveguides.

  • 6.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    Schwarz-Christoffel Mappings for Nonpolygonal Regions2008In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 31, no 1, p. 94-111Article in journal (Refereed)
    Abstract [en]

    An approximate conformal mapping for an arbitrary region Ω bounded by a smooth curve Γ is constructed using the Schwarz–Christoffel mapping for a polygonal region in which Ω is embedded. An algorithm for finding this so-called outer polygon is presented. The resulting function is a conformal mapping from the upper half-plane or the unit disk to a region R, approximately equal to Ω. R is bounded by a C∞ curve, and since the mapping function originates from the Schwarz–Christoffel mapping and tangent polygons are used to determine it, important properties of Γ such as direction, linear asymptotes, and inflexion points are preserved in the boundary of R. The method makes extensive use of existing Schwarz–Christoffel software in both the determination of outer polygons and the calculation of function values. By the use suggested here, the capabilities of such well-written software are extended.

  • 7.
    Andersson, Anders
    Jönköping University, School of Engineering, JTH, Mathematics.
    Using a zipper algorithm to find a conformal map for a channel with smooth boundary2006In: Mathematical Modeling of Wave Phenomena: 2nd Conference, 2006, p. 378-Conference paper (Refereed)
    Abstract [en]

    The so called geodesic algorithm, which is one of the zipper algorithms for conformal mappings, is combined with a Schwarz–Christoffel mapping, in its original or in a modified form, to produce a conformal mapping function between the upper half-plane and an arbitrary channel with smooth boundary and parallel walls at the end.

  • 8.
    Andersson, Anders
    et al.
    Jönköping University, School of Engineering, JTH, Mathematics.
    Nilsson, Börje
    International Centre for Mathematical modelling, Växjö University.
    Acoustic Transmission in Ducts of Various Shapes with an Impedance Condition2008In: International Conference on Numerical Analysis and Applied Mathematics 2008, Melville: American Institute of Physics , 2008, p. 33-36Conference paper (Refereed)
    Abstract [en]

    Propagation of acoustic waves in a two-dimensional duct with an impedance condition at the boundary, is studied. The duct is assumed to have two ends at infinity being asymptotically straight, but otherwise to be arbitrarily shaped.The so called Building Block Method allows us to synthesize propagation properties for ducts with complicated geometries from results for simpler ducts. Conformal mappings can be used to transform these simple ducts to straight ducts with constant cross-sections.By using recently developed techniques for numerical conformal mappings, it is possible to construct a transformation between an infinite strip and an arbitrarily shaped duct with smooth or piecewise smooth boundary, keeping both smoothness and the well controlled boundary direction towards infinity that the above mentioned method requires.To accomplish a stable formulation of the problem, we express it in terms of scattering operators. The resulting differential equation is solved using wave splitting and invariant embedding techniques. We expand the involved functions in Fourier series, and hence, it is possible to give the operators a matrix representation. Numerical results are produced using truncated matrices.

  • 9.
    Belov, I.
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology.
    Alavizadeh, Z.
    Lindgren, M.
    Application of engineering optimization to evaluate heating-based humidity management in electronics enclosures2011In: Electronic Environment Konferens & Mässa 2011, Stockholm Älvsjö,5-6 April 2011, Invited speech, 2011, p. 161-162Conference paper (Other academic)
  • 10.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Anti-Moisture Methodology for Electronics Enclosures in Harsh Storage Environments2008Conference paper (Other academic)
  • 11.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Evaluation of Anti-Moisture Measures in Electronics Enclosure: Application of CFD Modelling2009Conference paper (Other academic)
  • 12.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Humidity Management in Electronics Enclosure under Severe Climatic Conditions2008Conference paper (Other academic)
  • 13.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Optimization of PCB Heater Heating Profile for Power-efficient Humidity Management in Electronics Enclosures: OPTIMUS Worldwide User Meeting 2010, Antwerp, Belgium, November 22-23, 20102010Other (Other academic)
  • 14.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Bespalov, Michail
    Klochkova, Ludmila
    Kuleshov, Alexander
    Suzan, Dmitriy
    Tishkin, Vladimir
    Транспортная модель распространения газообразных примесей в атмосфере города = Transport model of gas impurities spread processes in urban area2000In: Математическое моделирование: (Mathematical Modelling), ISSN 0234-0879, Vol. 12, no 11, p. 38-46Article in journal (Refereed)
  • 15.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Bespalov, Michail
    Klochkova, Ludmila
    Pavlova, Natalya
    Suzan, Dmitriy
    Tishkin, Vladimir
    Сравнение моделей распространения загрязнений в атмосфере = Comparative Analysis of models of pollutions spreading in atmosphere1999In: Математическое моделирование: (Mathematical Modelling), ISSN 0234-0879, Vol. 11, no 8, p. 52-64Article in journal (Refereed)
  • 16.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Lindgren, Mats
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Bergner, Fredrik
    Bornoff, Robin
    CFD aided reflow oven profiling for PCB preheating in a soldering process2007In: Proceedings of the 8th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Micro-Electronics and Micro-Systems: EuroSimE 2007, Piscataway, NJ.: IEEE , 2007, p. 535-542Conference paper (Refereed)
    Abstract [en]

    A CFD-aided reflow oven profile prediction algorithm has been developed and applied to modelling of preheating of a PCB with non-uniform distribution of component thermal mass in a forced air convection solder reflow oven. The iterative algorithm combines an analytic approach with CFD modelling. It requires an experimentally validated CFD model of the solder reflow oven and a CFD model of the PCB as main inputs. Results of computational experiments have been presented to reveal good agreement between predicted PCB profiles and corresponding CFD calculations. Application guidelines contained in the description of the algorithm will assist potential users both during the virtual prototyping phase of a PCB including designing for assembly and in the phase of reflow oven profiling.

  • 17.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Lindgren, Mats
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Bergner, Fredrik
    Bornoff, Robin
    CFD aided reflow oven profiling for PCB preheating in a soldering process: Part 1(2)2007In: Electronic Environment, no 3, p. 25-28Article in journal (Other academic)
  • 18.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Lindgren, Mats
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Bergner, Fredrik
    Bornoff, Robin
    CFD aided reflow oven profiling for PCB preheating in a soldering process: Part 2(2)2007In: Electronic Environment, no 4, p. 25-27Article in journal (Other academic)
  • 19.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Lindgren, Mats
    Ryden, Jan
    Alavizadeh, Zahra
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    CFD Assisted Design Evaluation and Experimental Verification of a Logic-Controlled Local PCB Heater for Humidity Management in Electronics Enclosure2010In: IEEE EuroSimE 2010, 26-28 April, Bordeaux, France, 2010Conference paper (Refereed)
    Abstract [en]

    Humidity management of commercial-of-the-shelf electronic components in non-controlled climatic environments can be realized e.g. by introducing a local printed circuit board heater. By choosing appropriate size and location of the heater plate in the vicinity of the critical electronic packages, and utilizing logic control function, it is possible to improve the quality of local humidity management and reduce power consumption of the heater, which is important especially in case of battery driven portable or vehicle mounted devices. A computational fluid dynamics assisted methodology has been developed to determine the best feasible design of the heater, followed by experimental verification of the constructed logic controlled heater. The experiment has been performed in a harsh climatic environment including temperature variation from +33°C to +40°C, and relative humidity variation from 54% to 80%. Analysis of the experimental %RH and temperature curves as well as power profile of the heater has confirmed the feasibility of the chosen approach to maintain greater than 9°C difference between the electronics package surface temperature and the local dew point temperature, by applying discrete power pulses with the amplitude less than 6 W.

  • 20.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Rydén, Jan
    Lindblom, Joakim
    Zhang, Yafan
    Hansson, T
    Bergner, Fredrik
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Application of CFD Modelling for Energy Efficient Humidity Mangement of an Electronics Enclosure in Storage under Severe Climatic Conditions2008In: International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems, 2008: EuroSimE 2008., IEEE , 2008, p. 430-437Conference paper (Refereed)
    Abstract [en]

    A CFD modelling methodology including experimental validation has been developed and applied for investigation of anti-moisture measures in a non- hermetic electronics enclosure containing a number of printed circuit boards, and placed in a severe storage environment. In the climatic test the temperature and the relative humidity have been varried from +33degC to +71degC and from 14% to 80%, respectively. Enclosure heater solutions have been parametrically studied by simulation. A heating strategy involving various power levels has been determined, which is just sufficient to maintain the internal relative humidity below 60%, thereby reducing the risk for dew formation on the electronics assembly.

  • 21.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Wingbrant, Helena
    Spetz, Anita-Lloyd
    Sundgren, Hans
    Thuner, Bo
    Svenningstorp, Henrick
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Thermal and flow analysis of SiC-based gas sensors for automotive applications2004In: Proceedings of the 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems, 2004: EuroSimE 2004., IEEE , 2004, p. 475-482Conference paper (Refereed)
    Abstract [en]

    Different block and tube mounting alternatives for SiC-based gas sensors were studied by means of temperature measurements and simulation of heat transfer and gas flow for steady state conditions. The most preferable tube mounting design was determined. Simulation-based guidelines were developed for designing tube-mounted gas sensors in the exhaust pipes of diesel and petrol engines, taking into account thermal constraints and flow conditions.

  • 22.
    Belov, Ilja
    et al.
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Wingbrant, Helena
    Spetz, Anita-Lloyd
    Sundgren, Hans
    Thuner, Bo
    Svenningstorp, Henrik
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    CFD analysis of packaging and mounting solutions for SiC-based gas sensors in automotive applications2006In: Sensor Letters, ISSN 1546-198X, Vol. 4, no 1, p. 29-37Article in journal (Refereed)
    Abstract [en]

    Simulation-based guidelines were developed for designing tube-mounted gas sensors in the exhaust pipes of diesel and petrol engines, taking into account thermal constraints and gas flow conditions. Different block and tube mounting alternatives for SiC-based gas sensors were studied by means of temperature measurements and simulation of steady state heat transfer and gas flow. Design variables included the number of fins in the heat sink mounted on the inlet tube, the inlet construction, the mounting tube orientation, and the micro-heater substrate placement inside the mounting tube. The most preferable tube mounting design was determined with respect to the thermal performance of the sensor structure and with respect to the gas flow parameters, which are important for the sensor's selectivity, sensitivity and response time.

  • 23.
    Brundin, Ethel
    et al.
    Jönköping University, Jönköping International Business School, JIBS, EMM (Entrepreneurship, Marketing, Management). Jönköping University, Jönköping International Business School, JIBS, Center for Family Enterprise and Ownership.
    Wigren, Caroline
    Jönköping University, Jönköping International Business School, JIBS, EMM (Entrepreneurship, Marketing, Management).
    Isaacs, Eslyn
    Friedrich, Chris
    Visser, Kobus
    Triple Helix Networks in a Multicultural Context: Triggers and Barriers for Fostering Growth and Sustainability2008In: Journal of Developmental Entrepreneurship, ISSN 1084-9467, Vol. 13, no 1, p. 77-98Article in journal (Refereed)
    Abstract [en]

    This article deals with Triple Helix (university, industry and government co-operation) from an institutional theory perspective. The empirical context is the Western Cape Region in South Africa and the focus is entrepreneurship development. The purpose is twofold: first, the existing Triple Helix model is adapted to the South African context; and second, facilities and impediments for working according to Triple Helix in South Africa are identified. The empirical material consists of a survey and three longitudinal case studies illustrating the degree of co-operation between the three parties. The article contributes to knowledge about how the Triple Helix model works on a regional level in a developing country. The study draws the following conclusions: when co-operation is to be identified between the three actors, only two of the three are involved; one missing link in the Triple Helix model is the focus on the entrepreneur; co-operation between the three parties are incidental rather than planned and there is lack of structure. In turn, some of these conclusions may be an effect of institutional changes on a national level. For a normative legacy, the article proposes a set of suggestions for incorporating all relevant parties on a practical level.

  • 24.
    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.

  • 25.
    Burman, Erik
    et al.
    University College London, London, United Kingdom.
    Elfverson, Daniel
    Umeå University, Umeå, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå University, Umeå, Sweden.
    Larsson, Karl
    Umeå University, Umeå, Sweden.
    Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed)
    Abstract [en]

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

  • 26.
    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)
    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.

  • 27.
    Burman, Erik
    et al.
    University College London, United Kingdom.
    Elfverson, Daniel
    Umeå universitet, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    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.

  • 28.
    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.

  • 29.
    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.

  • 30.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    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.

  • 31.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    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.

  • 32.
    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. 

  • 33.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    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. 

  • 34.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Augmented Lagrangian finite element methods for contact problems2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, no 1, p. 173-195Article in journal (Refereed)
    Abstract [en]

    We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution. 

  • 35.
    Burman, Erik
    et al.
    University College London, London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    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.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    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 domains2019In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 179, p. 728-736Article 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.

  • 37.
    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.

  • 38.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, no 6, p. 2247-2282Article in journal (Refereed)
    Abstract [en]

    We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in Rd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R3.

  • 39.
    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.

  • 40.
    Burman, Erik
    et al.
    Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Samvin, David
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    A cut finite element method for elliptic bulk problems with embedded surfaces2019In: GEM - International Journal on Geomathematics, ISSN 1869-2672, E-ISSN 1869-2680, Vol. 10, no 1, article id 10Article in journal (Refereed)
    Abstract [en]

    We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractures. In particular the Laplace–Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples. 

  • 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, Materials and Manufacturing.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    Zahedi, Sara
    The Royal Institute of Technology (KTH), Department of Mathematics, Stockholm, Sweden.
    Stabilized CutFEM for the convection problem on surfaces2019In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 103-139Article 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.
    Burman, Erik N.
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    A cut finite element method with boundary value correction for the incompressible Stokes equations2019In: Numerical mathematics and advanced applications ENUMATH 2017, Cham: Springer, 2019, Vol. 126, p. 183-192Conference paper (Refereed)
    Abstract [en]

    We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018). 

  • 45.
    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.

  • 46. Chedid, Michel
    et al.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Experimental Analysis and Modelling of Textile Transmission Line for Wearable Applications2007In: International Journal of Clothing Science and Technology, ISSN 0955-6222, E-ISSN 1758-5953, Vol. 19, no 1, p. 59-71Article in journal (Refereed)
    Abstract [en]

    Purpose – The paper seeks, by means of measurement and modelling, to evaluate frequency dependent per-unit-length parameters of conductive textile transmission line (CTTL) for wearable applications and to study deterioration of these parameters when CTTL is subjected to washing.

    Design/methodology/approach – The studied transmission line is made of Nickel/Copper (Ni/Cu) plated polyester ripstop fabric and is subjected to standard 60°C cycle in a commercial off-the-shelf washing machine. The per-unit-length parameters (resistance and inductance) and characteristic impedance of the line are extracted from measurements before and after washing. Using the measurement data an equivalent circuit is created to model the degradation of the line. The circuit is then integrated in a three-dimensional transmission line matrix (TLM) model of the transmission line.

    Findings – Both an electrical equivalent circuit and a TLM model are developed describing the degradation of the conductive textile when washed. A severe deterioration of the electrical parameters of the line is noticed. Experimental and modelling results are in good agreement in the addressed frequency band.

    Research limitations/implications – Analysis is performed for frequencies up to 10?MHz. The developed TLM model can be used to conduct parametric studies of the CTTL. To counteract the degradation of the line, protective coating is to be considered in further studies.

    Originality/value – This paper extends knowledge of the subject by experimental and simulation-based characterization of the CTTL when subjected to washing cycles.

  • 47.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering.
    A mixed method for elasticity with the curl of displacements as a drilling degree of freedomManuscript (preprint) (Other academic)
    Abstract [en]

    We present a mixed method for the linearized elasticity equations with independent approximation of the curl of the displacements.The curl can be seen as a drilling degree of freedom allowing for coupling with rotating objects and the direct application of moments of force.

  • 48.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Nonconforming rotated Q1 tetrahedral element with explicit time stepping for elastodynamics2012In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 91, no 10, p. 1105-1114Article in journal (Refereed)
    Abstract [en]

    In this paper, we apply a rotated bilinear tetrahedral element to elastodynamics in R3. This element performs superior to the constant strain element in bending and, unlike the conforming linear strain tetrahedron, allows for row-sum lumping of the mass matrix. We study the effect of different choices of approximation (point- wise continuity versus edge average continuity) as well as lumping versus consistent mass in the setting of eigenvibrations. We also use the element in combination with the leapfrog method for time domain com- putations and make numerical comparisons with the constant strain and linear strain tetrahedra. 

  • 49.
    Hansbo, Peter
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Burman, Erik
    Universiy College London.
    Claus, Susanne
    University College London.
    Larson, Mats G.
    Umeå University.
    Massing, André
    Simula Research Laboratory.
    CutFEM: Discretizing geometry and partial differential equations2015In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, p. 472-501Article in journal (Refereed)
    Abstract [en]

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

  • 50.
    Hansbo, Peter
    et al.
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Jonsson, Tobias
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    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.
    A Nitsche method for elliptic problems on composite surfaces2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 505-525Article in journal (Refereed)
    Abstract [en]

    We develop a finite element method for elliptic partial differential equations on so called composite surfaces that are built up out of a finite number of surfaces with boundaries that fit together nicely in the sense that the intersection between any two surfaces in the composite surface is either empty, a point, or a curve segment, called an interface curve. Note that several surfaces can intersect along the same interface curve. On the composite surface we consider a broken finite element space which consists of a continuous finite element space at each subsurface without continuity requirements across the interface curves. We derive a Nitsche type formulation in this general setting and by assuming only that a certain inverse inequality and an approximation property hold we can derive stability and error estimates in the case when the geometry is exactly represented. We discuss several different realizations, including so called cut meshes, of the method. Finally, we present numerical examples. 

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