Combining set propagation with finite element methods for time integration in transient solid mechanics problems

•Novel approach for time integration in heat transfer and structural dynamics.•Procedure based on reachability analysis techniques and finite element discretization.•Solution sets include all exact trajectories converging to the true reachable states.•Set-based initial conditions or loads can be han...

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Vydáno v:Computers & structures Ročník 259; s. 106699
Hlavní autoři: Forets, Marcelo, Freire Caporale, Daniel, Pérez Zerpa, Jorge M.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Elsevier Ltd 15.01.2022
Elsevier BV
Témata:
Algorithms
Discretization
Equations of motion
Finite element method
Heat transfer
Initial conditions
Mathematical models
Methods
Numerical integration
Numerical methods
Numerical verification
Partial differential equations
Propagation
Reachability analysis
Solid mechanics
Structural dynamics
Time integration
Finite element method
Structural dynamics
Reachability analysis
Numerical verification
Heat transfer
ISSN:0045-7949, 1879-2243
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Shrnutí:•Novel approach for time integration in heat transfer and structural dynamics.•Procedure based on reachability analysis techniques and finite element discretization.•Solution sets include all exact trajectories converging to the true reachable states.•Set-based initial conditions or loads can be handled by the approach.•An efficient open source toolkit is implemented using JuliaReach and ONSAS. The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear structural dynamics problems modeled with partial differential equations (PDEs). While different algorithms for direct integration of the equations of motion exist, exploring all feasible behaviors for varying loads, initial states and fluxes in models with large numbers of degrees of freedom remains a challenging task. In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis. Assuming a set of initial states and inputs, the proposed method consists in the construction of a union of sets (flowpipe) that enclose the infinite number of solutions of the spatially discretized PDE. We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods. We conclude that, for problems with single known initial conditions, the proposed method is accurate. For problems with uncertain initial conditions included in sets, the proposed method can compute all the solutions of the system more efficiently than numerical integration methods.
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ISSN:0045-7949
1879-2243
DOI:10.1016/j.compstruc.2021.106699