22 research outputs found
Smooth Particle Hydrodynamics Applied to Fracture Mechanics
No full textA numerical method commonly referred to as smooth particle hydrodynamics (SPH) is implemented in two dimensions for solid mechanics in general and fracture mechanics in particular. The implementation is tested against a few analytical cases: a vibrating plate, a bending plate, a modus I crack and a modus II crack. A conclusion of these tests is that a better way of treating a shortcoming of SPH called tensile instability is needed. A study is made on the best choice of a vital parameter called the smoothing radius, and it is found that a good choice of the smoothing radius is roughly 1.5 times the initial particle spacing
High Order Cut Finite Element Methods for Wave Equations
No full textThis thesis considers wave propagation problems solved using finite element methods where a boundary or interface of the domain is not aligned with the computational mesh. Such methods are usually referred to as cut or immersed methods. The motivation for using immersed methods for wave propagation comes largely from scattering problems when the geometry of the domain is not known a priori. For wave propagation problems, the amount of computational work per dispersion error is generally lower when using a high order method. For this reason, this thesis aims at studying high order immersed methods. Nitsche's method is a common way to assign boundary or interface conditions in immersed finite element methods. Here, penalty terms that are consistent with the boundary/interface conditions are added to the weak form. This requires that special quadrature rules are constructed on the intersected elements, which take the location of the immersed boundary/interface into account. A common problem for all immersed methods is small cuts occurring between the elements in the mesh and the computational domain. A suggested way to remedy this is to add terms penalizing jumps in normal derivatives over the faces of the intersected elements. Paper I and Paper II consider the acoustic wave equation, using first order elements in Paper I, and using higher order elements in Paper II. High order elements are then used for the elastic wave equation in Paper III. Papers I to III all use continuous Galerkin, Nitsche's method, and jump-stabilization. Paper IV compares the errors of this type of cut finite element method with two other numerical methods. One result from Paper II is that the added jump-stabilization results in a mass matrix with a high condition number. This motivates the investigation of alternatives. Paper V considers a hybridizable discontinuous Galerkin method. This paper investigates to what extent local time stepping in combination with cell-merging can be used to overcome the problem of small cuts
Towards higher order immersed finite elements for the wave equation
No full textWe consider solving the scalar wave equation using immersed finite elements. Such a method might be useful, for instance, in scattering problems when the geometry of the domain is not known a priori. For hyperbolic problems, the amount of computational work per dispersion error is generally lower when using higher order methods. This serves as motivation for considering a higher order immersed method. One problem in immersed methods is how to enforce boundary conditions. In the present work, boundary conditions are enforced weakly using Nitsche's method. This leads to a symmetric weak formulation, which is essential when solving the wave equation. Since the discrete system consists of symmetric matrices, having real eigenvalues, this ensures stability of the semi-discrete problem. In immersed methods, small intersections between the immersed domain and the elements of the background mesh make the system ill-conditioned. This ill-conditioning becomes increasingly worse when using higher order elements. Here, we consider resolving this issue using additional stabilization terms. These terms consist of jumps in higher order derivatives acting on the internal faces of the elements intersected by the boundary.eSSENC
Towards higher order immersed finite elements for the wave equation [Elektronisk resurs]
No full textWe consider solving the scalar wave equation using immersed finite elements. Such a method might be useful, for instance, in scattering problems when the geometry of the domain is not known a priori. For hyperbolic problems, the amount of computational work per dispersion error is generally lower when using higher order methods. This serves as motivation for considering a higher order immersed method.One problem in immersed methods is how to enforce boundary conditions. In the present work, boundary conditions are enforced weakly using Nitsche's method. This leads to a symmetric weak formulation, which is essential when solving the wave equation. Since the discrete system consists of symmetric matrices, having real eigenvalues, this ensures stability of the semi-discrete problem.In immersed methods, small intersections between the immersed domain and the elements of the background mesh make the system ill-conditioned. This ill-conditioning becomes increasingly worse when using higher order elements. Here, we consider resolving this issue using additional stabilization terms. These terms consist of jumps in higher order derivatives acting on the internal faces of the elements intersected by the boundary.</p
Stabilized Cut Discontinuous Galerkin Methods for Advection-Reaction Problems
No full textWe develop novel stabilized cut discontinuous Galerkin methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in R-d where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two- and three-dimensional test problems.</p
High-order cut finite elements for the elastic wave equation
No full textA high-order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface is allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche’s method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts
Utveckling av Androidapplikation för airsoft
No full textAn Android-application that is intended to be used during airsoftgames (a gamesimilar to paintball) has been developed. In this application, which is called TactiX,players can join different teams and see their team members on a map. Through theapplication players can interact with one another in a variety of ways. They can send text messages, give orders and send drawings to each other. Players can also make observations of enemy players on the map. The application has been developed as athesis work at a bachelor of science level at Sweden Connectivity in Kista. Both theapplication and the server communication software have been developed
Stabilized Cut Discontinuous Galerkin Methods for Advection-Reaction Problems
No full textWe develop novel stabilized cut discontinuous Galerkin methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in \BbbR d where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two- and three-dimensional test problems.Kempe Foundation
Swedish Research Counci
