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1
Content available remote Regularity of fundamental solutions for Lévy-type operators
Szczypkowski Karol
Bulletin of the Polish Academy of Sciences. Mathematics
|
2023
|
Vol. 71, no 2
169--191
EN
For a class of non-symmetric non-local Lévy-type operators Lκ, which include those of the form Lκf(x) := Rd(f(x + z) − f(x) − 1|z|<1⟨z, ∇f(x)⟩)κ(x, z)J(z) dz, e prove regularity of the fundamental solution pκ to the equation ∂t = Lκ.
2
Content available A general study of fundamental solutions in aniotropicthermoelastic media with mass diffusion and voids
Chawla Vijay, Kamboj Deepmala
International Journal of Applied Mechanics and Engineering
|
2020
|
Vol. 25, no. 4
22--41
EN
The present paper deals with the study of a fundamental solution in transversely isotropic thermoelastic media with mass diffusion and voids. For this purpose, a two-dimensional general solution in transversely isotropic thermoelastic media with mass diffusion and voids is derived first. On the basis of the obtained general solution, the fundamental solution for a steady point heat source on the surface of a semi-infinite transversely isotropic thermoelastic material with mass diffusion and voids is derived by nine newly introduced harmonic functions. The components of displacement, stress, temperature distribution, mass concentration and voids are expressed in terms of elementary functions and are convenient to use. From the present investigation, some special cases of interest are also deduced and compared with the previous results obtained, which prove the correctness of the present result.
3
Content available n-sided polygonal hybrid finite elements involving element boundary integrals only for anisotropic thermal analysis
Cao R. F., Zhao X. J., Lin W. Q., Wang H.
Archives of Mechanics
|
2020
|
Vol. 72, nr 2
109--137
EN
As a combination of the traditional finite element method and boundary element method, the n-sided polygonal hybrid finite element method with fundamental solution kernels, named as HFS-FEM, is thoroughly studied in this work for two-dimensional heat conduction in fully anisotropic media. In this approach, the unknown temperature field within the polygon is represented by the linear combination of anisotropic fundamental solutions of problem to achieve the local satisfaction of the related governing equations, but not the specific boundary conditions and the continuity conditions across the element boundary. To tackle such a shortcoming, the frame temperature field is independently defined on the entire boundary of the polygonal element by means of the conventional one-dimensional shape function interpolation. Subsequently, by the hybrid functional with the assumed intra- and inter-element temperature fields, the stiffness equation can be obtained including the line integrals along the element boundary only, whose dimension is reduced by one compared to the domain integrals in the traditional finite elements. This means that the higher computing efficiency is expected. Moreover, any shaped polygonal elements can be constructed in a unified form with the same fundamental solution kernels, including convex and non-convex polygonal elements, to provide greater flexibility in meshing effort for complex geometries. Besides, the element boundary integrals endow the method higher versatility with a non-conforming mesh in the pre-processing stage of the analysis over the traditional FEM. No modification to the HFS-FEM formulation is needed for the non-conforming mesh and the element containing hanging nodes is treated normally as the one with more nodes. Finally, the accuracy, convergence, computing efficiency, stability of non-convex element, and straightforward treatment of non-conforming discretization are discussed for the present n-sided polygonal hybrid finite elements by a few applications in the context of anisotropic heat conduction.
4
Content available remote Potential method in the theory of thermoelasticity for materials with triple voids
Svanadze M.
Archives of Mechanics
|
2019
|
Vol. 71, nr 2
113--136
EN
In the present paper the linear theory of thermoelasticity for isotropic and homogeneous solids with macro-, meso- and microporosity is considered. In this theory the independent variables are the displacement vector field, the changes of the volume fractions of pore networks and the variation of temperature. The fundamental solution of the system of steady vibrations equations is constructed explicitly by means of elementary functions. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations.
5
Content available remote Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure
Svanadze M.
Archives of Mechanics
|
2017
|
Vol. 69, nr 4-5
347--370
EN
The purpose of the present paper is to develop the classical potential method in the linear theory of thermoelasticity for materials with a double porosity structure based on the mechanics of materials with voids. The fundamental solution of the system of equations of steady vibrations is constructed explicitly by means of elementary functions and its basic properties are established. The Sommerfeld-Kupradze type radiation conditions are established. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method and the theory of singular integral equations.
6
Content available remote Static and free vibration analysis of thin plates of the curved edges by the boundary element method considering an alternative formulation of boundary conditions
Guminiak M.
Engineering Transactions
|
2016
|
Vol. 64, iss. 1
3--32
EN
A static and dynamic analysis of Kirchhoff plates is presented in this paper. The proposed approach avoids Kirchhoff forces at the plate corners and equivalent shear forces at a plate boundary. Two unknown variables are considered at the boundary element node. The governing integral equations are derived using Betti’s theorem. The rectilinear and curved boundary element of the constant type are used. The non-singular formulation of the boundary (static analysis) and boundary-domain (free vibration analysis) integral equations with one and two collocation points associated with a single constant boundary element located at a plate edge are presented. Additionally, the classic three-node isoparametric curved boundary elements are introduced in static analysis according to the non-singular approach. Static fundamental solution and B`ezine technique are applied to the free vibration analysis. To establish the plate inertial forces, a plate domain is divided into triangular or annular sub-domains associated with one suitable collocation point.
7
Content available remote Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity
Kumar R., Vohra R., Gorla M. G.
Archives of Mechanics
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2016
|
Vol. 68, nr 4
263--284
EN
This paper is concerned with micropolar thermoelastic materials which have a double porosity structure. The system of the equations of the assumed model is based on the equations of motion, equilibrated stress equations of motion and heat conduction equation for material with double porosity. The explicit expressions for the fundamental solution of the system of equations in the case of steady vibrations are presented. The desired solutions are obtained by the use of elementary functions. Some basic properties are also established.
8
Content available remote Analytical solution for beams with multipoint boundary conditions on two-parameter elastic foundations
Aslami M., Akimov P. A.
Archives of Civil and Mechanical Engineering
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2016
|
Vol. 16, no. 4
668--677
EN
An efficient analytical method is presented for the closed form solution of continuous beams on two-parameter elastic foundations. The general form of the governing equation is reduced to a system of first-order differential equations with constant coefficients. The system is then solved using Jordan form decomposition for the coefficient matrix and construction of the fundamental solution. Common types of boundary conditions (pinned and roller support, hinge connection, fixed and free end) can be applied to an arbitrary point on the beam. The method has a completely computer-oriented algorithm, computational stability, and optimal conditionality of the resultant system and is a powerful alternative to the analytical solution of beams with multipoint boundary conditions on one- or two-parameter elastic foundations. Examples with different types of loading, boundary conditions, and foundation are presented to verify the method.
9
Content available remote An alternative approach to initial stability analysis of Kirchhoff plates resting on internal supports by the boundary element method
Guminiak M.
Engineering Transactions
|
2015
|
Vol. 63, iss. 3
273--296
EN
An initial stability of Kirchhoff plates supported on boundary and resting on internal supports is analysed in this paper. The internal supports are understood to be part of a plate surface or a line belonging to the plate. The proposed approach avoids Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary. Two unknown and independent variables are always considered at a boundary element node depending on the type of a plate edge such as the shear force and bending moment for a clamped edge, and the shear force and angle of rotation in normal direction for a simply-supported edge. For a free edge, the deflection and angle of rotation in normal direction are considered as two independent variables with additional angle of rotation in tangent direction which depends on boundary deflections. The two governing integral equations are derived using Betti’s theorem. These equations have the form of boundary-domain integral equations. The constant type of boundary element is used. The singular and non-singular formulations of the boundary-domain integral equations with one and two collocation points associated with a single boundary element located slightly outside of a plate edge are presented. To establish a plate curvature by double differentiation of the basic boundary-domain integral equation, the plate domain is divided into rectangular subdomains associated with suitable collocation points. According to the alternative approach, a plate curvature is also established by considering three collocation points located in close proximity to each other along a line parallel to one of the two axes of global coordinate system and establishment of appropriate difference operators.
10
Content available remote Fundamental solution in elasto-thermodiffusive (ETNP) semiconductor materials
Kumar R., Kansal T.
Archives of Mechanics
|
2015
|
Vol. 67, nr 5
371--384
EN
In this paper, the fundamental solution of system of differential equations in the theory of elasto-thermodiffusive(ETNP) semiconductor materials in case of steady oscillations in terms of elementary functions is constructed. Some basic properties of the fundamental solution are also established.
11
Content available remote An Alternative Approach of Initial Stability Analysis of Kirchhoff Plates by the Boundary Element Method
Guminiak M.
Engineering Transactions
|
2014
|
Vol. 62, iss. 1
33--59
EN
An initial stability of Kirchhoff plates is analysed in the paper. Proposed approach avoids Kirchhoff forces at the plate corner and equivalent shear forces at a plate boundary. Two unknown variables are considered at the boundary element node. The governing integral equations are derived using Betti theorem. The integral equations have the form of boundary and domain integral equations. The constant type of boundary element are used. The singular and non-singular formulation of the boundary-domain integral equations with one and two collocation points associated with a single boundary element located at a plate edge are presented. To establish a plate curvature by double differentiation of basic boundary-domain integral equation, a plate domain is divided into rectangular sub-domains associated with suitable collocation points. A plate curvature can also be establish by considering three collocation points located in close proximity to each other along line pararel to one of the two axes of global coordinate system and establishment of appropriate differential operators.
12
Content available remote Fundamental Solution for the Plane Problem in Magnetothermoelastic Diffusion Media
Kumar R., Chawla V.
Computational Methods in Science and Technology
|
2013
|
Vol. 19, No. 4
195--207
EN
The aim of the present paper is to study the fundamental solution in orthotropic magneto- thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic magnetothermoelastic diffusion media is derived. On the basis of thegeneral solution, the fundamental solution for a steady point heat source in an infinite and a semiinfinite orthotropic magnetothermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, some special cases of interest are also deduced and compared with the previously obtained results. The resulting quantities are computed numerically for infinite and semi-infinite magnetothermoelastic material and presented graphically to depict the magnetic effect.
13
Content available remote A new hybrid finite element approach for three-dimensional elastic problems
Cao C., Qin Q. H., Yu A.
Archives of Mechanics
|
2012
|
Vol. 64, nr 3
261-261
EN
A new fundamental solution based finite element method (HFS-FEM) is presented for analyzing three-dimensional (3D) elastic problems with body forces in this paper. It begins with deriving formulations of 3D HFS-FEM for elastic problems without body force and then the body force term is handled by means of the method of particular solution and radial basis function approximation. In our analysis, the homogeneous solution is obtained using the proposed HFS-FEM and the particular solution associated with the body force is approximated by radial basis functions. Several standard tests and numerical examples are considered to assess the capability and performance of the proposed method and elements. It is found that, comparing with conventional FEM (ABAQUS), the proposed method can achieve higher accuracy and efficiency when same element meshes are used. It is also found that the elements associated with this method are not very sensitive to mesh distortion and can be employed for problems involving nearly incompressible materials. This new method seems to be promising to deal with problems involving generalized body force, complex geometry, stress concentration and multi-materials.
14
A singular and non-singular BEM analysis of thin plates considering modified boundary conditions
Guminiak M.
Foundations of Civil and Environmental Engineering
|
2011
|
No. 14
5--45
EN
A static analysis of Kirchhoff plates using the boundary element method is presented in the paper. In this approach, physical boundary conditions are imposed. The Bettie theorem is used to derive the boundary integral equation. The collocation version of the boundary element method is presented. Typical and simplified, curved constant boundary elements are introduced.
15
Content available remote The method of the fundamental solutions and its modifications for electromagnetic field problems
Chen C. S., Reutskiy S. Y., Rozov V. Y.
Computer Assisted Mechanics and Engineering Sciences
|
2009
|
Vol. 16, No. 1
21-33
EN
The paper presents the method of fundamental solutions (MFS) for solving electromagnetic problems. We compare the MFS with the method of boundary integral equations in solution of potential problems. We demonstrate the MFS techniąue together with the Lapiace transform in application to the problem of scattering of electromagnetic pulses. A modification of the MFS - the method of approximate fundamental solutions (MAFS) is also considered in the paper. The method is applied to axisymmetric field problems. Numerical examples justifying the methods are presented.
16
The application of fundamental solutions in static analysis of thin plates resting on the internal elastic support
Pawlak Z., Guminiak M.
Foundations of Civil and Environmental Engineering
|
2008
|
No. 11
67-96
EN
A static analysis of Kirchhoff plates rested on the elastic internal supports has been discussed in the paper. The Finite Strip Method and Boundary Element Method have been used as an engineering tool in the analysis. Suitable fundamental solutions are applied in these method. Using BEM modified approach, there is no need to introduce the Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary. Two unknown and independent variables are considered at the boundary element node. The collocation points are located slightly outside the plate boundary, hence the quasidiagonal integrals of fundamental functions are non-singular. The constant type of boundary element has been used. According to the finite strip method a continuous structure is divided into a set of identical elements simply supported on opposite edges. The unknowns are the deflections and the transverse slope amplitudes along the nodal lines. The difference equation formulation is applied to express the equilibrium conditions of the discrete system. This reduces the number of degrees of freedom to be analyzed. The solution of one equilibrium difference equation yields the fundamental function of the considered plate strip. The fundamental solution derived in this way, can be used to solve the static problem of finite plate in analogically as in the boundary element method for continuous systems.
17
The free vibrations analysis of plates resting on continuous internal supports by the boundary element method
Guminiak M., Sumelka W.
Foundations of Civil and Environmental Engineering
|
2007
|
No. 10
69-97
EN
A free vibration analysis of Kirchhoff plates resting on continuous internal supports has been presented in the paper. Using the proposed approach, there is no need to introduce Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary. Two unknown and independent variables are considered at the boundary element node. The Bcttie theorem has been used to create the boundary integral equation. The collocation version of boundary element method with "constant" type of elements has been presented. The source points are located slightly outside the plate boundary, hence the quasi-diagonal integrals of fundamental functions are non-singular.
18
The analysis of internally supported thin plates by the boundary element method. Part 2 - Free vibration analysis
Guminiak M., Sygulski R.
Foundations of Civil and Environmental Engineering
|
2007
|
No. 9
43-74
EN
A free vibration analysis of internally supported Kirchhoff plates has been presented in the paper. Using the proposed approach, there is no need to introduce Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary [26], [31], [32], [34]. Two unknown and independent variables have been considered at the boundary element node. The Bettie theorem has been used to derive the boundary integral equation. The collocation version of boundary element method with elements of "constant" type has been presented. The source points are located slightly outside the plate boundary, hence the quasi-diagonal integrals of fundamental functions are non-singular [27], [31], [32], [34].
19
The analysis of internally supported thin plates by the boundary element method. Part 1 - Static analysis
Guminiak M., Sygulski R.
Foundations of Civil and Environmental Engineering
|
2007
|
No. 9
17-41
EN
A static analysis of Kirchhoff plates rested on the column supports has been presented in the paper. Using the proposed approach, there is no need to introduce Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary [14], [18], [19], [22]. Two unknown and independent variables have been considered at the boundary element node. The boundary integral equation has been derived using the Bettie theorem. The collocation points are located slightly outside a plate boundary, hence the quasi-diagonal integrals of fundamental functions are non-singular [15], [18], [19], [22], The constant types boundary element have been used.
20
The analysis of internally supported thin plates by the boundary element method. Part 3 - Initial stability analysis
Guminiak M., Jankowiak T.
Foundations of Civil and Environmental Engineering
|
2007
|
No. 10
51-67
EN
An initial stability of internally supported Kirchhoff plates has been analysed in the paper. Using the proposed approach, there is no need to introduce Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary. Two unknown variables are considered at the boundary clement node. The boundary integral equation is derived using the Bettie theorem. The collocation points arc located slightly outside the plate boundary, hence the quasi-diagonal integrals of fundamental functions are non-singular. The constant type of boundary clement is used.
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