Browsing by Author "Rabczuk, Timon"

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An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications
(2020) Samaniego Alvarado, Esteban Patricio; Anitescu, Cosmin; Goswami, Somdatta; Nguyen Thanh, Vien Minh; Hongwei, Guo; Hamdia, Khader M.; Zhuang, Xiaoying; Rabczuk, Timon
Partial Differential Equations (PDEs) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behavior of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best-known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate on applications that have an interest for Computational Mechanics. Most contributions explore this possibility have adopted a collocation strategy. In this work, we concentrate on mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. In order to prove the concepts, we deal with several problems and explore the capabilities of the method for applications in engineering.
Discontinuous modelling of shear bands using adaptive meshfree methods
(2008-01-15) Samaniego, Esteban; Rabczuk, Timon
A simple methodology to model shear bands as strong displacement discontinuities in an adaptive meshfree method is presented. The shear band is represented by a displacement jump at discrete particle positions. The displacement jump in normal direction is suppressed with penalty method. Loss of material stability is used as transition criterion from continuum to discontinuum. The method is twoand three-dimensional. Examples of complicated shear banding including transition from brittle-to-ductile failure are studied and compared to experimental data and other examples from the literature.
Isogeometric analysis of insoluble surfactant spreading on a thin film
(2020) Medina, David; Valizadeh, Navid; Samaniego Alvarado, Esteban Patricio; Jerves, Alex X; Rabczuk, Timon
Abstract In this paper we tackle the problem of surfactant spreading on a thin liquid film in the framework of isogeometric analysis. We consider a mathematical model that describes this phenomenon as an initial boundary value problem (IBVP) that includes two coupled fourth order partial differential equations (PDEs), one for the film height and one for the surfactant concentration. In order to solve this problem numerically, it is customary to transform it into a mixed problem that includes at most second order PDEs. However, the higher-order continuity of the approximation functions in Isogeometric Analysis (IGA) allows us to deal with the weak form of the fourth order PDEs directly, without the need of resorting to mixed methods. We demonstrate numerically that the IGA solution is able to reproduce results obtained before with mixed approaches. Complex phenomena such as Marangoni-driven fingering instabilities triggered by perturbations are easily captured.