Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind

Polishchuk, Olexandr

doi:10.23939/mmc2018.01.074

Cited by 4 publications

(2 citation statements)

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“…In NURBS based shape optimization, see [8,9], the NURBS basis functions are used to represent the geometry in the design model, and also as basis functions in the analysis model (for example in an isogeometric study), see [10][11][12]. In [13], Song et al consider the NURBS weights, in addition to the locations of the control points, as optimization variables.…”

Section: Ziani Mmentioning

confidence: 99%

Towards adaptation of the NURBS weights in shape optimization

Ziani¹

2022

Math. Model. Comput.

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Bézier based parametrisations in shape optimization have the drawback of using high degree polynomials to draw more complex shapes. To overcome this drawback, Non-Uniform Rational B-Splines (NURBS) are usually used. But, by considering the NURBS weights, in addition to the locations of the control points, as optimization variables, the dimension of the problem greatly increases and this would make the optimization process stiffer. In this work we propose, then, an algorithm to adapt the weights of NURBS in the parametrization of shape optimization problems. Unlike the coordinates of the control points, the weights are not considered, in this case, as variables of the optimization process. From the knowledge of an approximate optimal shape, we consider the set of all NURBS parametrizations of the same degree that approximate the shape in the sense of least squares. Then, we elect the parametrization associated with the most regular control polygon (least length of the control polygon). Numerical results show that the adaptive parametrization improves the performance of the optimization process.

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Section: Ziani Mmentioning

confidence: 99%

Towards adaptation of the NURBS weights in shape optimization

Ziani¹

2022

Math. Model. Comput.

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“…But, in [7], a more practical method was used to construct quadrature formulas for simple-layer and double-layer potentials, and in [8], the quadrature formulas for the normal derivative of simple-layer potential have been constructed and the error estimates for the constructed quadrature formulas have been obtained. Despite the successes in the field of numerical solution of integral equations of the first kind (see [2]- [5], [12]), the approximate solution of Neumann boundary value problems for the Helmholtz equation in two-dimensional space has not yet been studied by the method of integral equations of the first kind. This work is just about it.…”

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confidence: 99%