A fictitious domain method using equilibrated basis functions for harmonic and bi-harmonic problems in physics

Noormohammadi, Nima; Boroomand, B.

doi:10.1016/j.jcp.2014.04.011

Cited by 26 publications

(4 citation statements)

References 38 publications

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“…In this second approach, the complex domain is embedded into a larger, regular domain and the boundary conditions are approximated by a variety of different techniques. Examples include the adaptive fast multipole accelerated Poisson solver (e.g., [4]), which combines boundary and volume integral methods in the larger domain, fictitious domain methods (e.g., [5,6,7,8]) where Lagrange multipliers are applied in order to enforce the boundary conditions, immersed boundary (e.g., [9,10,11,12]), front-tracking (e.g., [13,14,15]) and arbitrary Lagrangian-Eulerian methods (e.g., [16,17,18,19]) utilize separate surface and volume meshes where force distributions are interpolated from the surface to the volume meshes, in a neighborhood of the domain boundary, to approximate the boundary conditions. In addition, a number of specialized methods have been designed to achieve better than first order accuracy in the L ∞ norm.…”

Section: Introductionmentioning

confidence: 99%

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Guo

Lowengrub

2020

Journal of Computational Physics

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The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain and source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher-order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in , the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, ∝ h, the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in , e.g., O( 2 ) in the L 2 norm and O( p ) with 1.5 ≤ p ≤ 2 in the L ∞ norm. Our analytic results are confirmed numerically in stationary and moving domains where the level set method is used to capture the dynamics of the domain boundary and to construct the smoothed characteristic function.

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Section: Introductionmentioning

confidence: 99%

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Guo

Lowengrub

2020

Journal of Computational Physics

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“…This may be performed by finding the so‐called equilibrated‐based functions, as shown in the works of Boroomand and Noormohammadi. () This is the issue on which we currently focus on.…”

Section: Discussionmentioning

confidence: 99%

A robust time‐space formulation for large‐scale scalar wave problems using exponential basis functions

Movahedian

Boroomand

Mansouri

2018

Numerical Meth Engineering

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Summary In this paper, a new effective boundary node method is presented for the solution of acoustic problems, directly in time domain, using exponential basis functions. Unlike many other methods using boundary information, the final coefficient matrix is sparse. The formulation is well suited for domains whose extent is relatively larger than the distance traveled by the acoustic wave in an increment of time. The exponential basis functions used satisfy the time‐space governing equation. This helps to choose a relatively large time increment and a moderate number of boundary points, which leads to reduction of computation time. The computation is performed incrementally using a weighted residual in time. Through a series of numerical examples, it is shown that the method, when combined with a domain decomposition strategy, is effectively capable of solving various 1‐ to 3‐dimensional acoustic problems.

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“…Also Gaussian exponential functions in the normalized space are used as weight functions, W is a weighting parameter controlling the sharpness of the weight function, which we consider equal to 30. A comprehensive study on this parameter may be found in Noormohammadi and Boroomand (2014). The weight points l , k are regularly distributed in n w1 rows and n w2 columns over normalized Ω 0 so that, with n w1 and n w2 as, In the case of variable coefficients for PDE, as a result of heterogeneous materials, the coefficients should be replaced by a set of incomplete monomials selected from the Pascal triangle.…”

Section: Homogeneous Solutionmentioning

confidence: 99%

Equilibrated Basis Functions for Static Analysis of In-plane Heterogeneous Laminated Composite Plates in Boundary and Meshfree Approaches

Azizpooryan

Noormohammadi

Boroomand

2021

Iran J Sci Technol Trans Mech Eng

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In this paper, static analysis of in-plane heterogeneous laminated composite plates is studied using equilibrated basis functions in a meshfree style based on first order shear deformation theory. The governing equations are satisfied through a weighted residual integration over a fictitious rectangular domain submerging the main domain of the plate. The integration is decomposed into a set of 1D pre-evaluated ones, which removes the need for numerical integration during the solution progress and considerably reduces the computational costs. The resulted bases may be implemented both in a boundary form and in a meshless local approach. Proper correlation between adjacent nodes in the meshless form ensures complete continuity of both the deformation and the stress components throughout the plate domain. Boundary conditions are applied independently of the equilibrium satisfaction through a simple collocation technique, which drastically decreases the required effort. The achieved results, compared with those presented in the literature or by commercial codes, reveal proper accuracy and convergence of the method. A comparative study on the effect of in-plane heterogeneity of the stiffness coefficients is also presented after verification of the method.

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A fictitious domain method using equilibrated basis functions for harmonic and bi-harmonic problems in physics

Cited by 26 publications

References 38 publications

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

A robust time‐space formulation for large‐scale scalar wave problems using exponential basis functions

Equilibrated Basis Functions for Static Analysis of In-plane Heterogeneous Laminated Composite Plates in Boundary and Meshfree Approaches

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