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Concept# Computer algebra system

Summary

A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.
Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics.
General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as:
*a user interface allowin

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Accuracy is critical if we are to trust simulation predictions. In settings such as ﬂuid- structure interaction it is all the more important to obtain reliable results to understand, for example, the impact of pathologies on blood ﬂows in the cardiovascular system. In this paper, we propose a computational strategy for simulating ﬂuid structure interaction using high order methods in space and time. First, we present the mathematical and computational core framework, Life, underlying our multi-physics solvers. Life is a versatile library allowing for 1D, 2D and 3D partial diﬀerential solves using h/p type Galerkin methods. Then, we brieﬂy describe the handling of high order geometry and the structure solver. Next we outline the high-order space- time approximation of the incompressible Navier-Stokes equations and comment on the algebraic system and the preconditioning strategy. Finally, we present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the ﬂuid-structure interaction problem as well as some initial results.

Related courses (4)

CS-550: Formal verification

We introduce formal verification as an approach for developing highly reliable systems. Formal verification finds proofs that computer systems work under all relevant scenarios. We will learn how to use formal verification tools and explain the theory and the practice behind them.

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The aim of this course is to provide the background in scientific computing. The class includes a brief introduction to basic programming in c++, it then focus on object oriented programming and c++ specific programming techniques.

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This course provides practical experience in the numerical simulation of fluid flows. Numerical methods are presented in the framework of the finite volume method. A simple solver is developed with Matlab, and a commercial software is used for more complex problems.

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs.
In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation.
The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system.
We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency.
We apply our strategy to a broad range of problems described by parametrized PDEs:
(i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations.
Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results.
Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques.
Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling.

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Dynamic fragmentation is a fast and catastrophic failure that takes place when an extreme load is applied to a material. During this process complex crack patterns develop. Cracks initiate at internal micro-defects, they propagate, eventually branch, and finally coalesce by forming fragments. This complex phenomenon was initially mainly studied with experimental and analytical methods, with particular emphasis on fragment sizes and shapes distributions. However, in the last decades the rapid development of computer capabilities led to an increasing number of numerical studies. In this work a numerical analysis of dynamic fragmentation is presented. Simulations are based on the finite-element method with dynamic insertion of cohesive elements. This method lets elastic waves spread unaltered and at the same time lets cracks initiate, propagate, branch and coalesce freely at any element borders. In order to use fine 3D meshes with several hundreds thousands of elements, all the algorithms have been coded in parallel. The presented numerical analysis constitutes one of the first applications of this new method in the fragmentation modeling of brittle materials. The studied model consists of a hollow sphere of aluminum oxide subjected to uniform radial expansion. Material defects are reproduced by imposing random stress thresholds around the model following a Weibull distribution. Several strain rates and radial thicknesses are considered, and statistics about fragment sizes and shapes are computed. For a small thickness, fragmentation is similar to that of a 2D plate in expansion. However, in thicker spheres, cracks can branch and merge also in the radial direction. They ultimately develop more complex patterns, transitioning to a mixed 2D/3D regime or pure 3D regime.

2014