scholarly journals DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Computation ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 75
Author(s):  
Valentin Alekseev ◽  
Maria Vasilyeva ◽  
Uygulaana Kalachikova ◽  
Eric T. Chung
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Homogeneous Boundary Condition ◽  
Multiscale Model ◽  
System Size ◽  
Numerical Results ◽  
Basis Functions ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Grid Construction

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

scholarly journals Solusi Persamaan keseimbangan Massa Reaktor Menggunakan Metode Pemisahan Variabel

CAUCHY ◽  
2015 ◽  
Vol 4 (1) ◽  
pp. 41
Author(s):  
Mohammad Syaiful Arif ◽  
Mohammad Jamhuri
Keyword(s):  
Boundary Condition ◽  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Mass Balance ◽  
Closed System ◽  
Mass Concentration ◽  
Separation Of Variables ◽  
Homogeneous Boundary Condition ◽  
Mass Coefficient ◽  
Reactor Equation

Mass balance of reactor equation express the change of mass concentration of substances in and out of the closed system. This equation has inhomogeneous boundary conditions, that is the conditions at the time of its entry to the reactor and the conditions under which the substance out of the reactor. In this study, the mass concentration of substances produced after the reaction in the reactor is zero. In the inhomogeneous boundary conditions, using the method of separation of variables, there are obstacles to complete the equation. So we need to first transformation. Transformation is done with the aim to change the conditions which originally inhomogeneous boundary into a homogeneous boundary condition, so the method of separation of variables can be used to solve partial differential equations that have a homogeneous boundary conditions. The results obtained by the analysis, the faster a substance that spreads to the reactor, the less amount of mass concentration of substances that undergo a change; the greater the mass coefficient of substances that react in the reactor, the more the number of mass concentration of substances that are subject to change

A SLOPING BOUNDARY CONDITION FOR EFFICIENT PE CALCULATIONS IN RANGE–DEPENDENT ACOUSTIC MEDIA

1996 ◽  
Vol 04 (01) ◽  
pp. 11-27 ◽  
Author(s):  
GARY H. BROOKE ◽  
DAVID J. THOMSON ◽  
PHILIP M. WORT
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Boundary Conditions ◽  
Standard Test ◽  
Numerical Results ◽  
Normal Displacement ◽  
Test Cases ◽  
Approximate Form ◽  
Acoustic Media ◽  
Horizontal Interface

The traditional one-way parabolic equation (PE) formulation for range-dependent layered acoustic media is modified to include effects associated with the boundary conditions along a sloping interface. Essentially, the boundary condition for continuity of the normal displacement along a sloping interface is cast in an approximate form which does not depend on range but does contain terms up to second order in the derivatives with respect to depth. The new sloping-boundary condition is then applied along an "equivalent" horizontal interface within each range-independent step of the PE. Numerical results obtained for standard test cases indicate that the sloping-boundary condition, incorporated into a one-way PE, maintains the efficiency yet improves the accuracy of forward predictions.

Homogenization of singularly perturbed Dirichlet problems in perforated domains

2000 ◽  
Vol 130 (1) ◽  
pp. 35-51
Author(s):  
Viêt Há Hoáng
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behaviour ◽  
Error Estimates ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Perforated Domain ◽  
Dirichlet Problems ◽  
Perforated Domains

We study the singularly perturbed problem —εαΔuε + uε = f (α > 0) with the Dirichlet boundary condition in a perforated domain, in which the holes are distributed periodically with period 2ε throughout a fixed domain Ω. The asymptotic behaviour of uε when ε → 0, together with corrector results and error estimates in L2(Ω), are deduced for all sizes of holes. The behaviour of uε in is obtained for the cases where the size of holes is of order ε or is of a sufficiently smaller order. When the holes' size is of a sufficiently small order, as expected, uε has similar behaviour to that in the case of a non-varying domain.

Drag on spheres in micropolar fluids with non-zero boundary conditions for microrotations

2007 ◽  
Vol 590 ◽  
pp. 319-330 ◽  
Author(s):  
KARL-HEINZ HOFFMANN ◽  
DAVID MARX ◽  
NIKOLAI D. BOTKIN
Keyword(s):  
Boundary Condition ◽  
Velocity Field ◽  
Boundary Conditions ◽  
Rotation Rate ◽  
Constant Velocity ◽  
Homogeneous Boundary Condition ◽  
Rotation Vector ◽  
Resistance Force ◽  
Micropolar Fluids ◽  
Stokes Formula

The Stokes formula for the resistance force exerted on a sphere moving with constant velocity in a fluid is extended to the case of micropolar fluids. A non-homogeneous boundary condition for the micro-rotation vector is used: the micro-rotation on the boundary of the sphere is assumed proportional to the rotation rate of the velocity field on the boundary.

scholarly journals H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

2021 ◽  
Author(s):  
Hamid Haddadou
Keyword(s):  
Boundary Condition ◽  
Quasilinear Equation ◽  
Neumann Boundary ◽  
Real Variable ◽  
Perforated Domain ◽  
Periodic Case ◽  
Perforated Domains ◽  
Periodicity Condition ◽  
The Family ◽  
Quasilinear Problem

AbstractIn this paper, we aim to study the asymptotic behavior (when $$\varepsilon \;\rightarrow \; 0$$ ε → 0 ) of the solution of a quasilinear problem of the form $$-\mathrm{{div}}\;(A^{\varepsilon }(\cdot ,u^{\varepsilon }) \nabla u^{\varepsilon })=f$$ - div ( A ε ( · , u ε ) ∇ u ε ) = f given in a perforated domain $$\Omega \backslash T_{\varepsilon }$$ Ω \ T ε with a Neumann boundary condition on the holes $$T_{\varepsilon }$$ T ε and a Dirichlet boundary condition on $$\partial \Omega $$ ∂ Ω . We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices $$(x,d)\mapsto A^{\varepsilon }(x,d)$$ ( x , d ) ↦ A ε ( x , d ) is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d, the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to $$A^{\varepsilon }(\cdot ,d)$$ A ε ( · , d ) in the perforated domain. Once the $$H^{0}$$ H 0 -limit $$A^{0}(\cdot ,d)$$ A 0 ( · , d ) of the pair $$(A^{\varepsilon },T^{\varepsilon })$$ ( A ε , T ε ) is determined, in the second step, we deduce that the solution $$u^{\varepsilon }$$ u ε converges in some sense to the unique solution $$u^{0}$$ u 0 in $$H^{1}_{0}(\Omega )$$ H 0 1 ( Ω ) of the quasilinear equation $$-\mathrm{{div}}\;(A^{0}(\cdot ,u^{0})\nabla u )=\chi ^{0}f$$ - div ( A 0 ( · , u 0 ) ∇ u ) = χ 0 f (where $$ \chi ^{0}$$ χ 0 is $$L^{\infty }$$ L ∞ weak $$^{\star }$$ ⋆ limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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