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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

We consider solving the surface Helmholtz equation 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. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition hk<C, where h denotes the mesh size, k the wave number and C a constant depending mainly on the surface curvature κ, but not on the surface/mesh intersection. Optimal error estimates in the H1 and L2-norms follow.

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.

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. 

We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.

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.

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.

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.

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.

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.

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.

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. 

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. 

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. 

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.

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.

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.

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.

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.

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. 

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.

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.

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. 

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). 

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.

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.

Majoriteten av elever saknar idag förståelsen för hur de ska addera bråktal med hjälp av varierande metoder trots flera års undervisning i bråkaritmetik. Studien syftar till att generera en fördjupad förståelse av de tidigare identifierade kritiska aspekter som årskurs 4 elever möter vid additionsberäkningar av tal i bråkform. Studien har två huvudsakliga frågeställningar: 1) Hur synliggörs tidigare identifierade kritiska aspekter inom additionsberäkningar med tal i bråkform i årskurs 4 elevers lösningar och resonemang? 2) Vilka kritiska aspekter framkommer i elevers lösningar med numeriska och bildliga representationsformer? 

Studien omfattar en kombination av kvantitativa och kvalitativa metoder där inslag av variationsteorin integrerades. De tillämpade metoderna innefattar användning av arbetsblad samt genomförande av semi-strukturerade intervjuer. Totalt deltog 35 elever i undersökningen, varav 6 elever valdes ut för intervju. Resultaten av studien identifierade tre kritiska aspekter. Kritisk aspekt 1: Att urskilja hur additionsberäkning med tal i bråkform fungerar i jämförelse med hur heltal beräknas. Kritisk aspekt 2: Att urskilja hur additionsberäkning, med tal i bråkform, utförs med olika nämnare. Kritisk aspekt 3: Att urskilja hur additionsberäkning utförs med hjälp av representation av tal i bråkform som en del av en helhet. Dessa kritiska aspekter som framkommit i tidigare forskning bekräftar och stödjer resultaten från studien.

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RTk \times Qk, k \geq 0. Here Qk is the space of discontinuous polynomial functions of degree less than or equal to k and RT is the Raviart-Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on RTk \times Qk, k \geq 0. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs RT0 \times Q0, RT1 \times Q1, and BDM1 \times Q0 (where BDM is the Brezzi-Douglas-Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh.