First Steps in the Space Separated Representation of Models Defined in Complex Domains

2012 ◽  
Author(s):  
Chady Ghnatios ◽  
Amine Ammar ◽  
Alain Cimetiere ◽  
Aziz Hamdouni ◽  
Adrien Leygue ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Penalty Method ◽  
3D Models ◽  
Geometrical Parameters ◽  
Regular Domain ◽  
Inverse Technique ◽  
One Dimensional ◽  
Separated Representations ◽  
Separated Representation ◽  
Alternative Routes

Separated representations allow substituting the resolution of 3D models by a sequence of three one-dimensional problems [1]. This route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist. In a former work [2], we explored the route of immersing the domain into a hexahedral domain, and then use a kind of penalty method to solve the model whilst enforcing the boundary conditions. In the present work, we are analyzing two alternative routes. The first one consists of using an inverse technique in order to compute the boundary conditions on the border of the hexahedra in which the complex domain is immersed. The second one consists in solving the model in a regular domain for a number of geometrical parameters considered as extra-coordinates from which we could have access to the solution in any geometry resulting from a choice of those parameters.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

scholarly journals Pair-correlation ansatz for the ground state of interacting bosons in an arbitrary one-dimensional potential

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Przemysław Kościk ◽  
Arkadiusz Kuroś ◽  
Adam Pieprzycki ◽  
Tomasz Sowiński
Keyword(s):  
Boundary Conditions ◽  
Ground State ◽  
Pair Correlation ◽  
Contact Interactions ◽  
One Dimensional ◽  
Particle Correlations ◽  
Variational Scheme ◽  
Full Control ◽  
Arbitrary Shapes

AbstractWe derive and describe a very accurate variational scheme for the ground state of the system of a few ultra-cold bosons confined in one-dimensional traps of arbitrary shapes. It is based on assumption that all inter-particle correlations have two-body nature. By construction, the proposed ansatz is exact in the noninteracting limit, exactly encodes boundary conditions forced by contact interactions, and gives full control on accuracy in the limit of infinite repulsions. We show its efficiency in a whole range of intermediate interactions for different external potentials. Our results manifest that for generic non-parabolic potentials mutual correlations forced by interactions cannot be captured by distance-dependent functions.

scholarly journals Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements

2021 ◽  
Vol 11 (2) ◽  
pp. 609
Author(s):  
Tadeusz Chyży ◽  
Monika Mackiewicz
Keyword(s):  
Finite Elements ◽  
Strain Field ◽  
Main Idea ◽  
Deformation Field ◽  
Geometrical Parameters ◽  
Single Element ◽  
One Dimensional ◽  
New Type ◽  
Self Adaptation ◽  
Adaptation Method

The conception of special finite elements called multi-area elements for the analysis of structures with different stiffness areas has been presented in the paper. A new type of finite element has been determined in order to perform analyses and calculations of heterogeneous, multi-coherent, and layered structures using fewer finite elements and it provides proper accuracy of the results. The main advantage of the presented special multi-area elements is the possibility that areas of the structure with different stiffness and geometrical parameters can be described by single element integrated in subdivisions (sub-areas). The formulation of such elements has been presented with the example of one-dimensional elements. The main idea of developed elements is the assumption that the deformation field inside the element is dependent on its geometry and stiffness distribution. The deformation field can be changed and adjusted during the calculation process that is why such elements can be treated as self-adaptive. The application of the self-adaptation method on strain field should simplify the analysis of complex non-linear problems and increase their accuracy. In order to confirm the correctness of the established assumptions, comparative analyses have been carried out and potential areas of application have been indicated.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

scholarly journals Order parameter profiles in a system with Neumann – Neumann boundary conditions

2018 ◽  
Vol 145 ◽  
pp. 01009 ◽  
Author(s):  
Vassil M. Vassilev ◽  
Daniel M. Dantchev ◽  
Peter A. Djondjorov
Keyword(s):  
Boundary Conditions ◽  
Order Parameter ◽  
Neumann Boundary ◽  
Elliptic Functions ◽  
Thermodynamic System ◽  
Neumann Boundary Conditions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
The One ◽  
Field Plane

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

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