| 1. |
- Donatelli, M., et al.
(författare)
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Multigrid methods for cubic spline solution of two point (and 2D) boundary value problems
- 2016
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Ingår i: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274 .- 1873-5460. ; 104, s. 15-29
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose a scheme based on cubic splines for the solution of the second order two point boundary value problems. The solution of the algebraic system is computed by using optimized multigrid methods. In particular the transformation of the stiffness matrix essentially in a block Toeplitz matrix and its spectral analysis allow to choose smoothers able to reduce error components related to the various frequencies and to obtain an optimal method. The main advantages of our strategy can be listed as follows: (i) a fourth order of accuracy combined with a quadratic conditioning matrix, (ii) a resulting matrix structure whose eigenvalues can be compactly described by a symbol (this information is the key for designing an optimal multigrid method). Finally, some numerics that confirm the predicted behavior of the method are presented and a discussion on the two dimensional case is given, together with few 2D numerical experiments. (C) 2014 IMACS. Published by Elsevier B.V. All rights reserved.
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| 2. |
- Nguyen, Quoc Khanh, 1993-, et al.
(författare)
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Spectral analysis of the finite element matrices approximating 2D linearly elastic structures and multigrid proposals
- 2022
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Ingår i: Numerical Linear Algebra with Applications. - : John Wiley & Sons. - 1070-5325 .- 1099-1506.
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Tidskriftsartikel (refereegranskat)abstract
- Topology optimization aims to find the best material layout subject to given constraints. The so-called material distribution methods cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. When solving the governing equation using the finite element method, a large number of elements are used to discretize the design domain, and an element-wise constant function approximates the coefficient field in the considered design domain. This article presents a spectral analysis of the (large) coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field. Based on the spectral information, we design a multigrid method which turns out to be optimal, in the sense that the (arithmetic) cost for solving the related linear systems, up to a fixed desired accuracy, is proportional to the matrix-vector cost, which is linear in the corresponding matrix size. The method is tested, and the numerical results are very satisfactory in terms of linear cost and number of iterations, which is bounded by a constant independent of the matrix size.
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| 3. |
- Nguyen, Quoc Khanh, 1993-, et al.
(författare)
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On using a zero lower bound on the physical density in material distribution topology optimization
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 359
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Tidskriftsartikel (refereegranskat)abstract
- The current paper studies the possibility of allowing a zero lower bound on the physical density in material distribution based topology optimization. We limit our attention to the standard test problem of minimizing the compliance of a linearly elastic structure subject to a constant forcing. First order tensor product Finite Elements discretize the problem. An elementwise constant material indicator function defines the discretized, elementwise constant, physical density by using filtering and penalization. To alleviate the ill-conditioning of the stiffness matrix, due to the variation of the elementwise constant physical density, we precondition the system. We provide a specific spectral analysis for large matrix sizes for the one-dimensional problem with Dirichlet-Neumann conditions in detail, even if most of the mathematical tools apply also in a d-dimensional setting, d >= 2. It is easy to find an elementwise constant material indicator function so that the resulting preconditioned system matrix is singular when allowing the vanishing physical densities. However, for a large class of material indicator functions, the corresponding preconditioned system matrix has a condition number of the same order as the system matrix for the case when the physical density is one in all elements. Finally, we critically report and illustrate results from numerical experiments: as a conclusion, it is indeed possible to solve large-scale topology optimization problems, allowing a vanishing physical density, without encountering ill-conditioned system matrices during the optimization.
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| 4. |
- Nguyen, Quoc Khanh, 1993-, et al.
(författare)
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Spectral Analysis of the Finite Element Matrices Approximating 3D Linearly Elastic Structures and Multigrid Proposals
- 2022
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Ingår i: Mathematical and Computational Applications. - : MDPI. - 1300-686X .- 2297-8747. ; 27:5, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- The so-called material distribution methods for topology optimization cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. In practice, the finite element method is typically used to approximate that kind of governing equations by using a large number of elements to discretize the design domain, and an element-wise constant function approximates the coefficient field in that domain. This paper presents a spectral analysis of the coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field in three spatial dimensions. The given theoretical analysis is used for designing and studying an optimal multigrid method in the sense that the (arithmetic) cost for solving the problem, up to a fixed desired accuracy, is linear in the corresponding matrix size. Few selected numerical examples are presented and discussed in connection with the theoretical findings.
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| 5. |
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| 7. |
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| 8. |
- Aceto, Lidia, et al.
(författare)
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Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis
- 2020
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 57
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Tidskriftsartikel (refereegranskat)abstract
- In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.
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| 9. |
- Adriani, Andrea, et al.
(författare)
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Asymptotic Spectra of Large (Grid) Graphs with a Uniform Local Structure (Part I) : Theory
- 2020
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Ingår i: Milan Journal of Mathematics. - : Springer Science and Business Media LLC. - 1424-9286 .- 1424-9294. ; 88:2, s. 409-454
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Tidskriftsartikel (refereegranskat)abstract
- We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain omega subset of Rd, d >= 1. When omega=[0,1] , such graphs include the standard Toeplitz graphs and, for omega=[0,1](d), the considered class includesd-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides many information on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type.Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes. Nevertheless, more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks
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| 10. |
- Adriani, Andrea, et al.
(författare)
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Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II : Numerical applications
- 2024
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 437
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Tidskriftsartikel (refereegranskat)abstract
- In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Ω ⊂ Rd , d ≥ 1. When Ω = [0, 1], such graphs include the standard Toeplitz graphs and, for Ω = [0,1]d, the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.
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