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236 results on '"Gianmarco Manzini"'

1. A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

Author

Yu Leng, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad, and Jeeyeon Plohr

Subjects
fracture mechanics, high-order phase field models, virtual element method, arbitrary-order approximations, Applied mathematics. Quantitative methods, T57-57.97
Abstract

We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by [1,2,3], we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use $ C^0 $ and $ C^1 $ virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.

Published
2023
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2. Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

Author

Boian Alexandrov, Gianmarco Manzini, Erik W. Skau, Phan Minh Duc Truong, and Radoslav G. Vuchov

Subjects
multidimensional numerical integration, highly-dimensional cubatures, Gauss-Legendre integration rule, Clenshaw-Curtis integration rule, Mathematics, QA1-939
Abstract

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.

Published
2023
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48. VEM and the Mesh.

Author

Tommaso Sorgente, Daniele Prada, Daniela Cabiddu, Silvia Biasotti, Giuseppe Patanè 0001, Micol Pennacchio, Silvia Bertoluzza, Gianmarco Manzini, and Michela Spagnuolo

Published
2021

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