Details
| Original language | English |
|---|---|
| Article number | 103176 |
| Journal | Probabilistic Engineering Mechanics |
| Volume | 67 |
| Early online date | 3 Nov 2021 |
| Publication status | Published - Jan 2022 |
Abstract
This work presents a numerical Galerkin scheme based on discontinuous Legendre polynomials (DLPG) to solve the integral eigenvalue problem known as Fredholm integral equation of a second kind which is mainly used for random fields representation by means of Karhunen–Loève expansion. The main advantages of the proposed method are the simple applicability of constructing the Legendre bases over each local elemental domain without considering any continuity between the elements, in addition to the orthogonality properties of the Legendre polynomials. The latter result in a fast, easy assembly and sparse representation of functions forming the linear system of equations, unlike the conventional Galerkin methods which tend to be computationally demanding. Three covariance functions are examined to demonstrate the feasibility and accuracy of the proposed method. The convergence properties of the approximated eigenvalues and the second moment of the numerically approximated random fields using the DLPG approach are confirmed with h- and p-refinement regarding a one-dimensional example. Furthermore, the applicability and efficiency is shown for two-dimensional domains considering both, rectangular and arbitrary domain shapes.
Keywords
- Discontinuous Legendre polynomial based Galerkin method, Fredholm integral equation, Karhunen–Loève expansion, Random field discretisation
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Engineering(all)
- Civil and Structural Engineering
- Energy(all)
- Nuclear Energy and Engineering
- Engineering(all)
- Aerospace Engineering
- Physics and Astronomy(all)
- Condensed Matter Physics
- Engineering(all)
- Ocean Engineering
- Engineering(all)
- Mechanical Engineering
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In: Probabilistic Engineering Mechanics, Vol. 67, 103176, 01.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Implementation of Karhunen–Loève expansion using discontinuous Legendre polynomial based Galerkin approach
AU - Basmaji, A. A.
AU - Dannert, M. M.
AU - Nackenhorst, U.
PY - 2022/1
Y1 - 2022/1
N2 - This work presents a numerical Galerkin scheme based on discontinuous Legendre polynomials (DLPG) to solve the integral eigenvalue problem known as Fredholm integral equation of a second kind which is mainly used for random fields representation by means of Karhunen–Loève expansion. The main advantages of the proposed method are the simple applicability of constructing the Legendre bases over each local elemental domain without considering any continuity between the elements, in addition to the orthogonality properties of the Legendre polynomials. The latter result in a fast, easy assembly and sparse representation of functions forming the linear system of equations, unlike the conventional Galerkin methods which tend to be computationally demanding. Three covariance functions are examined to demonstrate the feasibility and accuracy of the proposed method. The convergence properties of the approximated eigenvalues and the second moment of the numerically approximated random fields using the DLPG approach are confirmed with h- and p-refinement regarding a one-dimensional example. Furthermore, the applicability and efficiency is shown for two-dimensional domains considering both, rectangular and arbitrary domain shapes.
AB - This work presents a numerical Galerkin scheme based on discontinuous Legendre polynomials (DLPG) to solve the integral eigenvalue problem known as Fredholm integral equation of a second kind which is mainly used for random fields representation by means of Karhunen–Loève expansion. The main advantages of the proposed method are the simple applicability of constructing the Legendre bases over each local elemental domain without considering any continuity between the elements, in addition to the orthogonality properties of the Legendre polynomials. The latter result in a fast, easy assembly and sparse representation of functions forming the linear system of equations, unlike the conventional Galerkin methods which tend to be computationally demanding. Three covariance functions are examined to demonstrate the feasibility and accuracy of the proposed method. The convergence properties of the approximated eigenvalues and the second moment of the numerically approximated random fields using the DLPG approach are confirmed with h- and p-refinement regarding a one-dimensional example. Furthermore, the applicability and efficiency is shown for two-dimensional domains considering both, rectangular and arbitrary domain shapes.
KW - Discontinuous Legendre polynomial based Galerkin method
KW - Fredholm integral equation
KW - Karhunen–Loève expansion
KW - Random field discretisation
UR - http://www.scopus.com/inward/record.url?scp=85119291087&partnerID=8YFLogxK
U2 - 10.1016/j.probengmech.2021.103176
DO - 10.1016/j.probengmech.2021.103176
M3 - Article
AN - SCOPUS:85119291087
VL - 67
JO - Probabilistic Engineering Mechanics
JF - Probabilistic Engineering Mechanics
SN - 0266-8920
M1 - 103176
ER -