Application of the meshless procedure for the elastoplastic torsion of prismatic rods

In this paper torsion of prismatic bars considering elastic-plastic material behavior is studied. Based on the Saint-Venant displacement assumption and the Romberg-Osgood model for the stress-strain relation, the boundary value problem for stress function is formulated. In reality an area of cross section of a bar has two regions: elastic with linear governing equation and plastic with non-linear governing equation. In the solution procedure, the meshless procedure based on the Homotopy Analysis Method HAM connected with the Method of Fundamental Solutions (MFS) and Radial Basis Functions (RBF) is applied. The considered nonlinear partial differential equation (PDE) is transform into a hierarchy of linear inhomogeneous PDEs. The accuracy of the obtained approximate solution is controlled by the number of components of the calculate solution, while the convergence of the process is monitored by an additional parameter of the method. The advantage of the proposed meshless approach is that it does not require the generation of a mesh on the domain or its boundary, but only using a cloud of arbitrary located nodes

Application of the method of fundamental solutions for inverse problems related to the determination of elasto-plastic properties of prizmatic bar

The problem of determining the elastoplastic properties of a prismatic bar from the given relation from experiment between torsional moment MT and angle of twist per unit of rod’s length θ is investigated as inverse problem. Proposed method of solution of inverse problem is based on solution of some sequences of direct problem with application of the Levenberg-Marquardt iteration method. In direct problem these properties are known and torsional moment as a function of angle of twist is calculated form solution of some non-linear boundary value problem. For solution of direct problem on each iteration step the method of fundamental solutions and method of particular solutions is used for prismatic cross section of rod. The non-linear torsion problem in plastic region is solved by means of the Picard iteration

Zastosowanie metody rozwiązań podstawowych i transformacji Laplace’a do odwrotnego niestacjonarnego problemu źródeł ciepła

The paper deals with the inverse determination of heat source in an unsteady heat conduction problem. The governing equation for the unsteady Fourier heat conduction in 2D region with unknown internal heat source is known as the inverse boundary-initial-value problem. The identification of strength of the heat source is achieved by using the boundary condition, initial condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of determination of the heat source, the Laplace transformation with the method of fundamental solution and radial basis functions is proposed. Due to ill conditioning of the inverse transient heat conduction problem, the Tikhonov regularization method based on SVD and L-curve criterion was used. As the test problems, the 2D inverse boundary-initial-value problems (2D IBIVP) in region with known analytical solutions are considered.Artykuł dotyczy odwrotnego problemu określenia mocy źródeł ciepła dla nieustalonego zagadnienia przewodzenia ciepła. Równanie różniczkowe nieustalonego przewodzenia ciepła Fouriera w obszarze dwuwymiarowym z nieznanymi wewnętrznymi źródłami ciepła jest znane jako odwrotny problem brzegowo-początkowy. Przy określeniu mocy źródeł ciepła korzysta się z warunku brzegowego, warunku początkowego oraz znanej wartości temperatury w wybranych punktach rozmieszczonych wewnątrz rozważanego obszaru. Do rozwiązania odwrotnego problemu źródeł ciepła została zaproponowana transformacja Laplace’a połączona z metodą rozwiązań podstawowych i promieniowymi funkcjami bazowymi. Ze względu na złe uwarunkowanie problemu, w pracy zastosowano metodę regularyzacji Tichonowa dla rozkładu SVD oraz kryterium L-krzywej. Jako przykłady testowe rozważono dwuwymiarowe odwrotne problemy brzegowo-początkowe (2D IBIVP) ze znanym rozwiązaniem analitycznymi

Application of the meshless procedure for the elastoplastic torsion of prismatic rods

In this paper torsion of prismatic bars considering elastic-plastic material behavior is studied. Based on the Saint-Venant displacement assumption and the Romberg-Osgood model for the stress-strain relation, the boundary value problem for stress function is formulated. In reality an area of cross section of a bar has two regions: elastic with linear governing equation and plastic with non-linear governing equation. In the solution procedure, the meshless procedure based on the Homotopy Analysis Method HAM connected with the Method of Fundamental Solutions (MFS) and Radial Basis Functions (RBF) is applied. The considered nonlinear partial differential equation (PDE) is transform into a hierarchy of linear inhomogeneous PDEs. The accuracy of the obtained approximate solution is controlled by the number of components of the calculate solution, while the convergence of the process is monitored by an additional parameter of the method. The advantage of the proposed meshless approach is that it does not require the generation of a mesh on the domain or its boundary, but only using a cloud of arbitrary located nodes

Odwrotne określenie objętościowego udziału włókien we wzmocnionym kompozycie z niedoskonałym kontaktem termicznym pomiędzy składnikami

The paper considers the problem of determination of the volume fraction of fibres in an unidirectionally reinforced composite in order to provide the appropriate effective thermal conductivity. The problem formulated in such a way should be treated as an inverse heat transfer problem. The thermal conductivities of constituents (fibres and matrix) and fibres arrangement are known. The calculations are carried out for an imperfect thermal contact between the fibres and matrix.W pracy rozważa się problem określenia objętościowego udziału włókien w jednokierunkowo wzmocnionym kompozycie w celu uzyskania odpowiedniego efektywnego współczynnika przewodzenia ciepła. Problem sformułowany w ten sposób jest traktowany jako odwrotny problem przewodzenia ciepła. Współczynniki przewodzenia ciepła składników (włókien i matrycy) oraz sposób ułożenia włókien są znane. Obliczenia są wykonane dla niedoskonałego kontaktu termicznego pomiędzy włóknami i matrycą

Application of the method of fundamental solutions for inverse problems related to the determination of elasto-plastic properties of prizmatic bar

The problem of determining the elastoplastic properties of a prismatic bar from the given relation from experiment between torsional moment MT and angle of twist per unit of rod’s length θ is investigated as inverse problem. Proposed method of solution of inverse problem is based on solution of some sequences of direct problem with application of the Levenberg-Marquardt iteration method. In direct problem these properties are known and torsional moment as a function of angle of twist is calculated form solution of some non-linear boundary value problem. For solution of direct problem on each iteration step the method of fundamental solutions and method of particular solutions is used for prismatic cross section of rod. The non-linear torsion problem in plastic region is solved by means of the Picard iteration