The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary va... more The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary value problem for the onedimensional heat conduction equation. The numerical experiments are directed at a short presentation of advantages of the interval solutions obtained in the floating-point interval arithmetic over the approximate ones. It is also shown how we can deal with errors that occur during computations in terms of interval analysis and interval arithmetic.
In the article we present an interval difference scheme for solving a general elliptic boundary v... more In the article we present an interval difference scheme for solving a general elliptic boundary value problem with Dirichlet’ boundary conditions. The obtained interval enclosure of the solution contains all possible numerical errors. A numerical example we present confirms that the exact solution belongs to the resulting interval enclosure.
The paper concerns the first approach to interval generalized finite differences. The conventiona... more The paper concerns the first approach to interval generalized finite differences. The conventional generalized finite differences are of special interest due to the fact that they can be applied to irregular grids (clouds) of points. Based on these finite differences we can compute approximate values of some derivatives (ordinary or partial). Furthermore, one can formulate the generalized finite difference method for solving some boundary value problems with a complicated boundary of a domain. The aim of this paper is to propose the interval counterparts of generalized finite differences. Under the appropriate assumptions the exact values of the derivatives are included in the interval values obtained.
Abstract A numerical method based on the method of fundamental solutions (MFS) and the method of ... more Abstract A numerical method based on the method of fundamental solutions (MFS) and the method of particular solutions (MPS) together with the successive-approximation iteration process is presented. The nonlinear behaviour of the material that hardens with plastic deformation is characterized by the Chakrabarty model. The considerations are based on the incremental theory of plasticity. Furthermore, the incremental strain equations relate the plastic strain increments to the total strains only (the stresses do not appear there). The method is used for solving three example boundary value problems that describe the stress state in some plates subjected to external loads. The accuracy of the results is examined on the basis of the boundary conditions fulfilment and the comparison with the finite element method (FEM). Finally, the regions of elastic/plastic deformation are identified. Then, the distribution of the equivalent stress is shown.
Abstract We apply a variable shape parameter Kansa–radial basis function (RBF) collocation method... more Abstract We apply a variable shape parameter Kansa–radial basis function (RBF) collocation method for the numerical solution of second and fourth order nonlinear boundary value problems in two dimensions. In the current approach, each RBF in the solution approximation is associated with a different shape parameter. These shape parameters are considered to be part of the unknowns along with the values of the coefficients of the RBFs in the solution approximation. The system of nonlinear equations resulting from the Kansa–RBF discretization is solved by directly applying a standard nonlinear solver. The proposed method is applied to several numerical examples.
We propose matrix decomposition algorithms for the efficient solution of the linear systems arisi... more We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.
An interval version of the conventional nine-point finite difference method for solving the two-d... more An interval version of the conventional nine-point finite difference method for solving the two-dimensional Laplace equation with the Dirichlet boundary conditions is proposed. This interval scheme is interesting due to the fact that the local truncation error of the conventional method is of the high (fourth) order, but it becomes of the sixth order for square mesh. In the theoretical approach presented, this error is bounded by some interval values and we can prove that the exact solution belongs to the interval solutions obtained.
International Journal of Solids and Structures, 2015
Abstract The plane elastoplastic problem for the stress state of a plate with a narrowing subject... more Abstract The plane elastoplastic problem for the stress state of a plate with a narrowing subjected to uniaxial extension is considered. The behaviour of the material that hardens with plastic deformation is characterised by the Ramberg–Osgood stress–strain relation. Since this relation provides a smooth continuous curve over the whole elastic and plastic deformation range, the same governing equation can be used for both deformation regions. The paper provides a method for solving the resulting nonlinear boundary-value problem. The algorithm is based on meshless methods, i.e. the method of fundamental solutions and the method of particular solutions, together with a Picard iteration process. The approximate solution, i.e. the stress function, obtained in each iteration step is a linear combination of fundamental and particular solutions. It can thus be further used to compute the values of stresses and some effective material parameters (i.e. the Young modulus and the Poisson ratio) at any point of the domain.
The paper concerns a problem of bioheat transfer between a single large blood vessel and a surrou... more The paper concerns a problem of bioheat transfer between a single large blood vessel and a surrounding tissue. On the basis of the conventional finite difference scheme with the appropriate truncation error terms included, the interval finite difference method is proposed. The interval values that contain the local truncation error of the conventional scheme can be approximated in the way described.
The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary va... more The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary value problem for the onedimensional heat conduction equation. The numerical experiments are directed at a short presentation of advantages of the interval solutions obtained in the floating-point interval arithmetic over the approximate ones. It is also shown how we can deal with errors that occur during computations in terms of interval analysis and interval arithmetic.
In the article we present an interval difference scheme for solving a general elliptic boundary v... more In the article we present an interval difference scheme for solving a general elliptic boundary value problem with Dirichlet’ boundary conditions. The obtained interval enclosure of the solution contains all possible numerical errors. A numerical example we present confirms that the exact solution belongs to the resulting interval enclosure.
The paper concerns the first approach to interval generalized finite differences. The conventiona... more The paper concerns the first approach to interval generalized finite differences. The conventional generalized finite differences are of special interest due to the fact that they can be applied to irregular grids (clouds) of points. Based on these finite differences we can compute approximate values of some derivatives (ordinary or partial). Furthermore, one can formulate the generalized finite difference method for solving some boundary value problems with a complicated boundary of a domain. The aim of this paper is to propose the interval counterparts of generalized finite differences. Under the appropriate assumptions the exact values of the derivatives are included in the interval values obtained.
Abstract A numerical method based on the method of fundamental solutions (MFS) and the method of ... more Abstract A numerical method based on the method of fundamental solutions (MFS) and the method of particular solutions (MPS) together with the successive-approximation iteration process is presented. The nonlinear behaviour of the material that hardens with plastic deformation is characterized by the Chakrabarty model. The considerations are based on the incremental theory of plasticity. Furthermore, the incremental strain equations relate the plastic strain increments to the total strains only (the stresses do not appear there). The method is used for solving three example boundary value problems that describe the stress state in some plates subjected to external loads. The accuracy of the results is examined on the basis of the boundary conditions fulfilment and the comparison with the finite element method (FEM). Finally, the regions of elastic/plastic deformation are identified. Then, the distribution of the equivalent stress is shown.
Abstract We apply a variable shape parameter Kansa–radial basis function (RBF) collocation method... more Abstract We apply a variable shape parameter Kansa–radial basis function (RBF) collocation method for the numerical solution of second and fourth order nonlinear boundary value problems in two dimensions. In the current approach, each RBF in the solution approximation is associated with a different shape parameter. These shape parameters are considered to be part of the unknowns along with the values of the coefficients of the RBFs in the solution approximation. The system of nonlinear equations resulting from the Kansa–RBF discretization is solved by directly applying a standard nonlinear solver. The proposed method is applied to several numerical examples.
We propose matrix decomposition algorithms for the efficient solution of the linear systems arisi... more We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.
An interval version of the conventional nine-point finite difference method for solving the two-d... more An interval version of the conventional nine-point finite difference method for solving the two-dimensional Laplace equation with the Dirichlet boundary conditions is proposed. This interval scheme is interesting due to the fact that the local truncation error of the conventional method is of the high (fourth) order, but it becomes of the sixth order for square mesh. In the theoretical approach presented, this error is bounded by some interval values and we can prove that the exact solution belongs to the interval solutions obtained.
International Journal of Solids and Structures, 2015
Abstract The plane elastoplastic problem for the stress state of a plate with a narrowing subject... more Abstract The plane elastoplastic problem for the stress state of a plate with a narrowing subjected to uniaxial extension is considered. The behaviour of the material that hardens with plastic deformation is characterised by the Ramberg–Osgood stress–strain relation. Since this relation provides a smooth continuous curve over the whole elastic and plastic deformation range, the same governing equation can be used for both deformation regions. The paper provides a method for solving the resulting nonlinear boundary-value problem. The algorithm is based on meshless methods, i.e. the method of fundamental solutions and the method of particular solutions, together with a Picard iteration process. The approximate solution, i.e. the stress function, obtained in each iteration step is a linear combination of fundamental and particular solutions. It can thus be further used to compute the values of stresses and some effective material parameters (i.e. the Young modulus and the Poisson ratio) at any point of the domain.
The paper concerns a problem of bioheat transfer between a single large blood vessel and a surrou... more The paper concerns a problem of bioheat transfer between a single large blood vessel and a surrounding tissue. On the basis of the conventional finite difference scheme with the appropriate truncation error terms included, the interval finite difference method is proposed. The interval values that contain the local truncation error of the conventional scheme can be approximated in the way described.
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Papers by Malgorzata Jankowska