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1

A meshless pseudospectral approach appliedto problems with weak discontinuities

Krowiak Artur

Computer Assisted Methods in Engineering and Science

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2022

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Vol. 29, no. 3

179--195

EN

In this paper, a meshless pseudospectral method is applied to solve problems possessingweak discontinuities on interfaces. To discretize a differential problem, a global inter-polation by radial basis functions is used with the collocation procedure. This leads toobtaining the differentiation matrix for the global approximation of the differential opera-tor from the analyzed equation. Using this matrix, the discretization of the problem isstraightforward. To deal with the differential equations with discontinuous coefficients onthe interface, the meshless pseudospectral formulation is used with the so-called subdo-main approach, where proper continuity conditions are used to obtain accurate results.In the present paper, the differentiation matrix for this method is derived and the choiceof a proper value of the shape parameter for radial functions in the context of the subdo-main approach is studied. The numerical tests show the effectiveness of the method whenusing regular or unstructured node distribution. They confirm that the approach preserveswell-known advantages of the radial basis function collocation method, i.e., rapid conver-gence, simplicity of the implementation and extends its usage for problems with weakdiscontinuity.

2

A meshless method for subsonic stall flutter analysis of turbomachinery 3D blade cascade

Prasad Chandra Shekhar, Šnábl Pavel, Pešek Luděk

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 6

art. no. e139000

EN

The analysis of subsonic stall flutter in turbomachinery blade cascade is carried out using a medium-fidelity reduced-order aeroelastic numerical model. The model is a type of field mesh-free approach and based on a hybrid boundary element method. The medium-fidelity flow solver is developed on the principle of viscous-inviscid two-way weak-coupling approach. The hybrid flow solver is employed to model separated flow and stall flutter in the 3D blade cascade at subsonic speed. The aerodynamic damping coefficient w.r.t. inter blade phase angle in traveling-wave mode is estimated along with other parameters. The same stability parameter is used to analyze the cascade flutter resistance regime. The estimated results are validated against experimental measurements as well as Navier-Stokes based high fidelity CFD model. The simulated results show good agreement with experimental data. Furthermore, the hybrid flow solver has managed to bring down the computational cost significantly as compared to mesh-based CFD models. Therefore, all the prime objectives of the research have been successfully achieved.

3

Selected Aspects of Meshless Method Optimization in the Room Acoustics with Impedance Boundary Conditions

Prędka Edyta, Kocan-Krawczyk Anna, Brański Adam

Archives of Acoustics

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2020

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Vol. 45, No. 4

647--654

EN

Two optimization aspects of the meshless method (MLM) based on nonsingular radial basis functions (RBFs) are considered in an acoustic indoor problem. The former is based on the minimization of the mean value of the relative error of the solution in the domain. The letter is based on the minimization of the relative error of the solution at the selected points in the domain. In both cases the optimization leads to the finding relations between physical parameters and the approximate solution parameters. The room acoustic field with uniform, impedance walls is considered. As results, the most effective Hardy’s Radial Basis Function (H-RBF) is pointed out and the numer of elements in the series solution as a function of frequency is indicated. Next, for H-RBF and fixed n, distributions of appropriate acoustic fields in the domain are compared. It is shown that both aspects of optimization improve the description of the acoustic field in the domain in a strictly defined sense.

4

Analysis of the Room Acoustic with Impedance Boundary Conditions in the Full Range of Acoustic Frequencies

Prędka Edyta, Brański Adam

Archives of Acoustics

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2020

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Vol. 45, No. 1

85--92

EN

An efficiency of the nonsingular meshless method (MLM) was analyzed in an acoustic indoor problem. The solution was assumed in the form of the series of radial bases functions (RBFs). Three representative kinds of RBF were chosen: the Hardy’s multiquadratic, inverse multiquadratic, Duchon’s functions. The room acoustic field with uniform, impedance walls was considered. To achieve the goal, relationships among physical parameters of the problem and parameters of the approximate solution were first found. Physical parameters constitute the sound absorption coefficient of the boundary and the frequency of acoustic vibrations. In turn, parameters of the solution are the kind of RBFs, the number of elements in the series of the solution and the number and distribution of influence points. Next, it was shown that the approximate acoustic field can be calculated using MLM with a priori error assumed. All approximate results, averaged over representative rectangular section of the room, were calculated and then compared to the corresponding accurate results. This way, it was proved that the MLM, based on RBFs, is efficient method in description of acoustic boundary problems with impedance boundary conditions and in all acoustic frequencies.

5

Nonsingular Meshless Method in an Acoustic Indoor Problem

Brański A., Prędka E.

Archives of Acoustics

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2018

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Vol. 43, No. 1

75--82

EN

An efficiency of the nonsingular meshless method is analyzed in an acoustic indoor problem. The solution is assumed in the form of the series of radial bases functions. The Hardy’s multiquadratic functions, as the bases, are taken into account. The room acoustic field with uniform, impedance walls is considered. The representative, rectangular cross section of the room is chosen. Practical combinations of acoustic boundary conditions, expressed through absorption coefficient values, are considered. The classical formulation of the boundary problem is used. It is established any coefficient in the multiquadratic functions depend on the number of influence points, the frequency and the absorption coefficient. All approximate results are calculated in relation to the exact ones. This way, it is proved that the meshless method based on the multiquadratic functions is simple and efficient method in the description of the complicated acoustic boundary problems for the low and medium ranges of frequency.

6

Analiza płaskiej fali naprężenia metodą ruchomych, najmniejszych kwadratów

Dornowski Wojciech

Modern Engineering

|

2017

|

1

1--17

PL

W pracy rozwiązano zadanie początkowo-brzegowe propagacji płaskiej fali naprężenia w tarczy stalowej metodą ruchomych najmniejszych kwadratów. Stosowano różne siatki węzłów. W przypadku siatek z węzłami generowanymi losowo istotne znaczenie ma wybór węzłów sąsiedztwa aproksymacyjnego. Przyjęto kryterium topologiczne wynikające z triangulacji zbioru węzłów, która minimalizuje łączną długość krawędzi triangulacji. Stwierdzono, że triangulacyjny sposób wyboru otoczenia aproksymacyjnego i wygładzanie siatki nieregularnej metodą Laplace’a znacznie poprawia dokładność rozwiązania. Efekt odbicia fali naprężenia od brzegu swobodnego modelowano przez wprowadzenia węzłów fikcyjnych (poza obszarem tarczy). Otrzymane rezultaty porównano z wynikami obliczeń metodą różnicową stwierdzając ich zgodność jakościową i ilościową.

EN

A meshless method based on the moving least squares aproximation is applied to stress wave propagation analysis. Two kinds of node meshes, the randomly generated mesh and the regular mesh are used. The nearest neighbors problem are developed from a triangulation that satisfies mini-mum edges length conditions. It is found that this method of neighbors choice significantly improves the solution accuracy. The reflection of stress waves from the free edge is model edusing fictitious nodes(outside the plate). The comparison with the finite difference results also demonstrated the accuracy of the proposed approach.

7

Development of simple effective cloud of nodes and triangular mesh generators for meshless and element-based analyses – implementation in Matlab

Milewski S.

Computer Assisted Methods in Engineering and Science

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2017

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Vol. 24, no. 3

157--180

EN

This paper is devoted to the development of the Matlab software dedicated to the generation of 2D arbitrarily irregular clouds of nodes and triangular meshes. They may be applied in numerical analyses of boundary value problems, based on both meshless and finite element discretization techniques, especially in the case of numerical homogenization in which the domain partitioning into disjoint subdomains may be required. Several Matlab functions are extended on the basis of the simple computational geometry-based ideas and concepts of engineering nature. A set of Matlab functions, attached to this paper, is discussed in detail, and examined on selected boundary value problems.

8

Zastosowanie radialnych funkcji bazowych do analizy pola akustycznego wnętrz pojazdów

Majkut L.

Autobusy : technika, eksploatacja, systemy transportowe

|

2016

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R. 17, nr 12

1183--1187

PL

W artykule opisano możliwości zastosowania jednej z tzw. metod bezsiatkowych do analizy pola akustycznego wnętrza pojazdu. Do analizy wybrano metodę kolokacyjną z bazą globalnych funkcji radialnych. W kolejnych punktach pracy opisano wykorzystywaną metodę analizy i jej zastosowanie do wyznaczenia częstości drgań własnych i charakterystyk amplitudowo-częstotliwościowych wnętrza przykładowego pojazdu. Charakterystyki wyznaczono w dwóch układach źródła: jedno źródło zlokalizowane na środku przedniej części pojazdu (układ siso) i dwa źródła umieszczone po bokach pojazdu (układ miso). Wartości częstości własnych porównano z analizą Metodą Elementów Skończonych.

EN

The work concerns the in acoustics of an vehicle interior with the Radial Based Functions. The collocation method was used for determination eigenvalues and frequency response functions of the car interior. All eigenvalues were compared with Finite Element Analysis results. All results indicate that using of multiquadric (MQ) RBF provide a results with very high accuracy in comparison to numerical results in acoustic analysis of 3D close domain.

9

Method of regularized sources for two-dimensional Stokes flow problems based on rational or exponential blobs

Wen S., Wang K., Zahoor R., Li M., Šarler B.

Computer Assisted Methods in Engineering and Science

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2015

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Vol. 22, no. 3

289--300

EN

The solution of Stokes flow problems with Dirichlet and Neumann boundary conditions is performed by a non-singular method of fundamental solutions (MFS) which does not require artificial boundary, i.e., source points of fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity is obtained from the analytical solution due to the action of the Dirac delta- type force. Instead of Dirac delta force, a non-singular function called blob, with a free parameter epsilon is employed, which is limited to Dirac delta function when epsilon is limited to zero. The analytical expressions for related Stokes flow pressure and velocity around such regularized sources have been derived for rational and exponential blobs in an ordered way. The solution of the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary and with their intensities chosen in such a way that the solution complies with the boundary conditions. A numerical example for two-dimensional (2D) driven cavity and a flow between parallel plates are chosen to assess the properties of the method. The results of the posed method of regularized sources (MRS have been compared with the results obtained by the fine-grid second-order classical finite difference method (FDM) and analytical solution. The results converge with finer discretisation; however, they depend on the value of epsilon. The method gives reasonably accurate results for the range of epsilon between 0.1 and 0.5 of the typical nodal distance on the boundary. Exponential blobs give slightly better results than the rational blobs; however, they require slightly more computing time. A robust and efficient strategy to find the optimal value of epsilon is needed in the perspective.

10

Analiza falowa tarczy w płaskim stanie naprężenia metodą ruchomych, najmniejszych kwadratów

Dornowski W.

Biuletyn Wojskowej Akademii Technicznej

|

2014

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Vol. 63, nr 3

189-206

PL

W pracy rozwiązano zadanie początkowo-brzegowe propagacji płaskiej fali naprężenia w tarczy stalowej metodą ruchomych najmniejszych kwadratów. Stosowano różne siatki węzłów. W przypadku siatek z węzłami generowanymi losowo istotne znaczenie ma wybór węzłów sąsiedztwa aproksymacyjnego. Przyjęto kryterium topologiczne wynikające z triangulacji zbioru węzłów, która minimalizuje łączną długość krawędzi triangulacji. Stwierdzono, że triangulacyjny sposób wyboru otoczenia aproksymacyjnego i wygładzanie siatki nieregularnej metodą Laplace’a znacznie poprawia dokładność rozwiązania. Efekt odbicia fali naprężenia od brzegu swobodnego modelowano przez wprowadzenie węzłów fikcyjnych (poza obszarem tarczy). Otrzymane rezultaty porównano z wynikami obliczeń metodą różnicową, stwierdzając ich zgodność jakościową i ilościową.

EN

A meshless method based on the moving least squares approximation is applied to stress wave propagation analysis. Two kinds of node meshes, the randomly generated mesh and the regular mesh are used. The nearest neighbours’ problem is developed from a triangulation that satisfies minimum edges length conditions. It is found that this method of neighbours’ choice significantly improves the solution accuracy. The reflection of stress waves from the free edge is modelled using fictitious nodes (outside the plate). The comparison with the finite difference results also demonstrated the accuracy of the proposed approach.

11

Properties of a Meshless Simulation Method of Diffusion from a Finite Source

Dawidowicz T.

Biuletyn Instytutu Systemów Informatycznych

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2011

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nr 8

25-33

EN

The paper presents an analysis of the properties of a numerical algorithm of a diffusion simulation from a finite source based on a meshless numerical method. The simulation time dependencies and simulation errors have been presented. The range of parameters were indicated for which the method achieves physical results.

PL

W pracy przedstawiono analizę właściwości algorytmu numerycznego symulacji dyfuzji ze źródła skończonego opartego o bezsiatkową metodę numeryczną. Wykazano zależności czasowe symulacji oraz przedstawiono błędy symulacji. Wskazano zakres parametrów, dla których metoda uzyskuje wyniki fizyczne.

12

Modelling of heat treatment of steel elements with the movement of coolant

Kulawik A., Bokota A.

Archives of Metallurgy and Materials

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2011

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Vol. 56, iss. 2

345-357

EN

A mathematical and numerical model of hardening process using the generalized finite difference method for the movement of fluid and heat transport have been proposed in this paper. To solve the Navier-Stokes equation the characteristic based split scheme (CBS) has been used. The solution of the heat transport equation with the convective term has been obtained by a stabilized meshless method. To determine of the phase transformation the macroscopic model built on the basis of CCT diagrams for continuous cooling of medium-carbon steel has been used. The temporary temperature fields, the phase transformation, thermal and structural strains for the heat treated element and the fields of temperature and velocity for the coolant have been determined. The comparative analysis of the results of calculations for the model without taking into account movement of coolant has been carried out. The effect of the notch in the shaft on the cooling rates and fields of the kinetics of the phase transformations has been presented.

PL

W pracy zaproponowano model matematyczny i numeryczny zjawisk termicznych oraz ruchów chłodziwa zbudowany z wykorzystaniem uogólnionej metody różnic skończonych. Do rozwiązania równania Naviera-Stokesa wykorzystano metodę rzutowania (CBS). Rozwiązanie równania przewodzenia ciepła z członem konwekcyjnym uzyskano na podstawie stabilizowanej bezsiatkowej metody różnic skończonych. Do modelowania przemian fazowych wykorzystano makroskopowy model zbudowany na podstawie analizy wykresów ciągłego chłodzenia CTPc dla stali średniowęglowej. Dla elementu obrabianego cieplnie określono chwilowe pola temperatury, udziały fazowe, odkształcenia termiczne, strukturalne oraz pala temperatury i prędkości cieczy chłodzącej. Przeprowadzono analizę porównawczą z wynikami obliczeń z ruchem i bez ruchu chłodziwa. W pracy przedstawiono także wpływ wycięcia (rowka na wałku) na pola prędkości chłodziwa oraz na kinetykę przemian fazowych.

13

Transient heat conduction by different versions of the Method of Fundamental Solutions – a comparison study

Kołodziej J. A. J. A., Mierzwiczak M.

Computer Assisted Mechanics and Engineering Sciences

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2010

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Vol. 17, no. 1

75-88

EN

The computational accuracy of three versions of the method of fundamental solutions (MFS) is compared. The first version of MFS is based on the Laplace transformation of the governing differential equations and of the boundary conditions. The second version of MFS is based on the fundamental solution of the governing differential equation and discretization in time. The third method approximates the temperature time derivative by finite difference scheme. As the test problems the 2D boundary-initial-value problems (2D_BIVP) in square rectangular region ? with known exact solutions are considered. Our numerical experiments show that all discussed methods achieve relatively accurate approximate solution but the third one offers less computational complexity and better efficiency.

14

Stacjonarne przewodzenie ciepła przez wielowarstwową ściankę o zależnym od temperatury współczynniku przewodzenia ciepła

Mierzwiczak M.

Zeszyty Naukowe Politechniki Poznańskiej. Budowa Maszyn i Zarządzanie Produkcją

|

2008

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Nr 9

67-79

PL

W artykule przedstawiono numeryczne rozwiązanie stacjonarnego zagadnienia przewodzenia ciepła przez wielowarstwową płaską i cylindryczną ściankę, której współczynnik przewodzenia zależy od temperatury. Do rozwiązania problemu zastosowano bezsiatkową metodę Kansy. Nieznane pole temperatury przyjmuje się w warstwach w postaci liniowej kombinacji radialnych funkcji bazowych (RBF). Kolokacyjne spełnienie równania rządzącego i warunków brzegowych prowadzi do nieliniowego układu równań rozwiązywanego metodą Newtona.

EN

This paper deals with numerical solution of heat transfer problem in multilayered plate and cylinder with temperature-dependent thermal conductivity. The Kansa meshless method was used for the solution of this problem. In this approach, the unknown's temperatures in layers are approximated by the radial basis functions, while the governing equation and the boundary conditions are imposed directly at the collocation points. The multiquadrics are used as the radial basis functions. Non-linear system of algebraic equations for coefficients at radial basis functions is solved by Newton method.

15

Trefftz radial basis functions (TRBF)

Kompiš V., Štiavnicky M., Žmindák M., Murčinková Z.

Computer Assisted Mechanics and Engineering Sciences

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2008

|

Vol. 15, No. 3/4

239-249

EN

The TRBF's are radial functions satisfying governing equation in the domain. They can be used as interpolation functions of the field variables especially in boundary methods. In present paper discrete dipoles are used to simulate composite material reinforced by stiff particles using with boundary point collocation method which does not require any meshing and any integration. The better the interpolation (unction satisfies also the boundary conditions, the more efficient it is. In examples it is shown that a triple dipole (which is a TRBK) located into the center of the particle can approximate the inter-domain boundary conditions very good, if the particles are not very close to each other and their size is not very different. In general problem the model can be used as very good start point for international improvements in refined model. (Composite reinforced by short fibres with very large aspect ratio continuous TRBF were developed. They enable to reduce problem considerably and to simulate complicated interactions for investigation such composites.

16

Optimal parameters of method of fundamental solutions for Poisson problems in heat transfer by means of genetic algorithms

Kołodziej J. A., Klekiel T.

Computer Assisted Mechanics and Engineering Sciences

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2008

|

Vol. 15, No. 2

99-112

EN

This paper describes the application of the method of fundamental solutions to the solution of the boundary value problems of the two-dimensional steady heat transfer with heat sources. For interpolation of an inhomogeneous term in Poisson equation the radial basis functions are used. Three cases of boundary value problems are solved and five cases of radial basis functions are used. For comparison purposes the boundary value problems for which exact solution exists were chosen. Application of method of fundamental solutions with boundary collocation and radial basis function for solution of inhomogeneous boundary value problems introduces some number of parameters related with these tools. For optimal choosing of these parameters the genetic algorithm is used. The results of numerical experiences related to optimal parameters are presented.

17

A comparison of meshless and finite element approaches to ductile damage in forming processes

Cesar de Sa J., Zheng C.

Computer Methods in Materials Science

|

2007

|

Vol. 7, No. 2

262-268

EN

In metal forming processes the damage associated with large deformations is a phenomenon that should be minimized or simply avoided as it usually leads to flawed parts. The initiation of plasticity and damage is caused by movement and accumulation of dislocations in metals but their nature and evolution is different. Ductile damage evolution in metals is usually associated with the initiation and growth of micro cracks and cavities, resulting in a progressive material softening. Damage growing influences indirectly the plastic behaviour by locally reducing the elementary area of resistance and therefore plasticity and damage should be coupled at the constitutive level. In the theory of Continuum Damage Mechanics the damage is represented by internal variables (of scalar, vectorial or tensor type) which give a measure of the deteriorated state at each representative volume of the material. This variable may then be used to define the effective stress state. Another important aspect is related with the fact that in ductile damage localization is similar to that associated with plastic strain. These physical phenomena are characterised by the accumulation of damage and large deformations within narrow bands. In experiments these localization zones display a finite width which may be related to the micro structure of the material. Classical theories of plasticity and damage mechanics, based on internal variable approaches, are local theories and do not include size effects associated to a characteristic dimension of the material. Their implementation in a finite element setting shows a pathologic effect of spatial mesh dependence because the constitutive models are unable to capture the limitation of the localization upon mesh refinement. In fact, the original hypothesis of homogeneous continuous models does not take into account large changes in the internal variables, like plastic strain and damage, in the localization zone. The aforementioned effect can be adequately explained by micro mechanical theories but their numerical implementation is still rather expensive. Non-local models have been proposed to bridge the gap between classical continuum theories and the micromechanical ones. In these models the evolution of some internal variables describing strain and damage in a specific point is also determined by the history of the surrounding material by including in the formulation averages or gradients of part or all of them. Some of theses models have proved to be effective when implemented in a finite element framework. Some claims have been made that the new class of computational methods, i.e. meshless methods, could be more effective when dealing with localization problems. Typically these new methods use a set of points and local support functions to represent the problem domain with no need of an additional mesh. This local support functions could then be broadened for the evaluation of the evolution of the internal variables, giving a non-local character to the solution. Therefore, in this work, an incursion was made into the application of these methods to this particular type of problems in order to investigate how meshless methods deal with ductile damage phenomena, if the unacceptable discretization dependence is also present and to assess how effectively the non-local and gradient models work in these settings. The chosen meshless method was the Reproducing Kernel Particle Method (RKPM). The material model was extended in order to include ductile damage effects by coupling the elastoplastic constitutive law with the damage evolution equations. Non-local and related gradient (explicit and implicit) models were also implemented using the RKPM. A set of numerical examples showed that the meshless solution scheme on ductile damage, exhibits the same type of dependence of solutions upon refinement of the geometrical discretization. Both implicit and explicit gradient and non-local models can alleviate this pathological behaviour. Nevertheless the explicit gradient model still presents a local behaviour by concentrating the damage on a narrower zone.

PL

Możliwość wykorzystania metody bez siatkowej Reproducing Kernel Particle Method (RKPM) do symulacji plastycznego pękania w procesach przeróbki plastycznej jest tematem niniejszej pracy. Zalety metody RKPM są porównane z konwencjonalnymi modelami MES, szczególnie pod względem problemów z dyskretyzacją badanego obszaru. Zastosowany model pękania bazuje na podejściu Lemaitre z uwzględnieniem rozgraniczenia pękania dla lokalnych obszarów rozciąganych i spęczanych. Zaimplementowane lokalne i globalne modele w formie jawnej i niejawnej są porównane i omówione w niniejszej pracy.

18

A meshless method for non-linear Poisson problems with high gradients

Algahtani H. I.

Computer Assisted Mechanics and Engineering Sciences

|

2006

|

Vol. 13, No. 3

367-377

EN

A meshless method for the solution of linear and non-linear Poisson-type problems involving high gradients is presented. The proposed method is based on collocation with 3rd order polynomial radial basis function coupled with the fundamental solution. The linear problem is solved by satisfying the boundary conditions and the governing differential equations over selected points over the boundary and inside the domain, respectively. In the case of the non-linear case, the resulted equations are highly non-linear and therefore, they are solved using an incremental-iterative procedure. The accuracy and efficiency of the method is verified through several numerical examples.

19

Application of meshless element free Galerkin method in two-dimensional heat conduction problems

Singh I. V., Sandeep K., Prakash R.

Computer Assisted Mechanics and Engineering Sciences

|

2004

|

Vol. 11, No. 4

265-274

EN

In this paper, meshless element free Galerkin method has been used to obtain the numerical solution of transient and steady state heat conduction problems in two-dimensional domains. The unknown function of temperature T(x) has been approximated by moving least square approximant Th(x). These approximants are constructed by using a weight function, a polynomial basis and a set of non-constant coefficients. Variational method is used to obtain the discrete equations. Essential boundary conditions are imposed by Lagrange multiplier technique. Two new weight functions namely hyperbolic and rational have been proposed. The results have been obtained for a two-dimensional model problem using different EFG weight functions and are compared with those obtained by finite element and analytical methods.

20

High-accuracy discretization methods for solid mechanics

Tolstykh A. I., Lipavskii M. V., Shirobokov D. A.

Archives of Mechanics

|

2003

|

Vol. 55, nr 5-6

531-553

EN

Novel high-accuracy computational techniques for solid mechanics problems are presented. They include fourth-order and arbitrary-order finite difference methods based on Pade-type differencing formulas and a meshless method which uses radial basis functions in a "finite difference'' mode. Some results illustrating high performance of the suggested numerical methods are displayed.