An Ultra Weak Finite Element Method as an Alternative to a Monte Carlo Method for an Elasto-Plastic Problem with Noise

Bensoussan, Alain; Mertz, Laurent; Pironneau, Olivier; Turi, János

doi:10.1137/090751141

Cited by 16 publications

(23 citation statements)

References 6 publications

(7 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To do so, we use probabilistic simulations, as explained bellow. In a similar manner to what was done in [BMPT09] and [FM12] to solve (1.2), here the solution (y n (t), z n (t)) of (1.5) has explicit formulae in each phase: either |z n (t)| ≤ Y , z n (t) > Y or z n (t) < −Y .…”

Section: Experiments On the Invariant Measure Using Probabilistic Simmentioning

confidence: 99%

“…In terms of engineering, this probability measure describes the probabilities that (1.1) experiences an elastic or a plastic state for large times, hence it is relevant for engineering purposes. Therefore, a numerical algorithm solving this probability measure has been proposed in [BMPT09] and then it has been applied to the estimation of the frequency of plastic deformation of (1.1) in [FM12].…”

Section: Background and Motivationsmentioning

confidence: 99%

See 1 more Smart Citation

Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise

Laurière

Mertz

2015

ESAIM: Proc.

Self Cite

View full text Add to dashboard Cite

To cite this version:Mathieu Laurière, Laurent Mertz. Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise. [ AMO, 2008, it has been shown that the solution of a stochastic variational inequality modeling an elasto-plastic oscillator excited by a white noise has a unique invariant probability measure. The latter is useful for engineering in order to evaluate statistics of plastic deformations for large times of a certain type of mechanical structure. However, in terms of mathematics, not much is known about its regularity properties. From then on, an interesting mathematical question is to determine them. Therefore, in order to investigate this question, we introduce in this paper approximate solutions of the stochastic variational inequality by a penalization method. The idea is simple: the inequality is replaced by an equation with a nonlinear additional term depending on a parameter n penalizing the solution whenever it goes beyond a prespecified area. In this context, the dynamics is smoother. In a first part, we show that the penalized process converges towards the original solution of the aforementioned inequality on any finite time interval as n goes to ∞. Then, in a second part, we justify that for each n it has at least one invariant probability measure. We conjecture that it is unique, but unfortunately we are not able to prove it. Finally, we provide numerical experiments in support of our conjecture. Moreover, we give an empirical convergence rate of the sequence of measures related to the penalized process. Résumé. Dans un travail récent de A.Bensoussan et J.Turi Degenerate Dirichlet Problems Relatedto the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008, il aété montré que la solution d'une inéquation variationnelle stochastique modélisant un oscillateurélasto-plastique excité par un bruit blanc admet une unique mesure de probabilité invariante. Cette dernière est utile en science de l'ingénieur pour estimer les statistiques des déformations plastiques en temps grands d'un certain type de structure mécanique. Dès lors, un problème mathématique intéressant est de déterminer la régularité de cette mesure. Afin d'étudier ce problème, nous introduisons ici des solutions approchées de l'inéquation par une méthode de pénalisation. Ainsi, l'inéquation est remplacée par uneéquation avec un terme nonlinéaire additionnel dépendant d'un certain paramètre n pénalisant la solution en dehors d'un domaine admissible. Dans ce contexte, la dynamique stochastique est plus régulière. Dans un premier temps, nous montrons la convergence lorsque n tend vers ∞ du processus pénalisé vers la solution de l'inéquation sur tout intervalle de temps fini. Puis dans un second temps, nous montrons que pour chaque n le processus pénalisé est dissipatif et qu'ainsi il admet au moins une mesure invariante. Malheureusement, bien qu'ayant quelques pistes, nous ne sommes pas (encore) capables de démontrer son unicité, que nous conjecturons. En contrepartie, nous l'étudions numériq...

show abstract

Section: Experiments On the Invariant Measure Using Probabilistic Simmentioning

confidence: 99%

Section: Background and Motivationsmentioning

confidence: 99%

Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise

Laurière

Mertz

2015

ESAIM: Proc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It should be pointed out that the presentation remains formal as there are mathematical subtleties behind the definition of these stopping times. An existence and uniqueness result in a weak sense (see for instance, definition 3.1 page 300 in [25]) for an equation similar to (5) can be found in the works of Jacob [23,24]. To be more precise, it concerns an integrated Wiener process constrained to stay in r0, 8q by a partially elastic boundary at 0.…”

Section: 2mentioning

confidence: 99%

A Feynman–Kac formula approach for computing expectations and threshold crossing probabilities of non-smooth stochastic dynamical systems

Mertz

Stadler

Wylie

2019

Physica D: Nonlinear Phenomena

Self Cite

View full text Add to dashboard Cite

We present a computational alternative to probabilistic simulations for nonsmooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman-Kac formula, where the underlying stochastic processes satisfy variational inequalities modelling elasto-plastic and obstacle oscillators. We then focus on solving them numerically. The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities.Here and in the remainder of the paper, µ, λ are positive numbers, b is a given threshold, T ą 0 is a given time, h ě 0 and f, g, ϕ, ψ are continuous functions. For cases in which H " 0 and pF, Gq satisfy appropriate smoothness conditions, a natural setting for characterizing such quantities with partial differential equations (PDEs) is to use the Feynman-Kac formula (FKf) [45]. Furthermore, if in addition F and G can be written in terms of a Hamiltonian structure Hpx, yq, in the sense F px, yq fi BH Bx px, yq, Gpx, yq fi BH By px, yq`ηpx, yq BH Bx px, yq, px, yq P R 2 ,where η is a well behaved function, then the process pX t , Y t q belongs to the class of Stochastic Hamiltonian Systems (SHS) [48] for which quantities of type C and C 1 are well defined. Applications that make use of this framework abound in many areas of science and engineering, e.g., finance, chemistry, biology, neuroscience, economy and mechanics. 6

show abstract

“…To this end, we apply the following averaging procedure. We first consider the ultra-weak variational formulation [44][45][46] that can be derived from (12) by integration by parts. Instead of shifting the derivative on the solution u h as in (13), the ultra-weak form shifts all derivatives to the test function ıu h .…”

Section: Symmetrization By Averaging With the Ultra-weak Formulationmentioning

confidence: 99%

“…In particular, the stiffness matrix of the weighted residual formulation is non-symmetric.If we choose approximation and test functions based on the same GLL basis in the sense of a Bubnov-Galerkin method, the symmetry of stiffness matrices in collocated FEA can be restored. To this end, we average the weighted residual formulation with a dual variational formulation based on the ultra-weak formulation [44][45][46]. Symmetry is essential for reducing memory and speeding up formation and assembly procedures and for the application of efficient iterative solvers based on conjugate gradient (CG) methods.…”

mentioning

confidence: 99%

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Schillinger

Evans

Frischmann

et al. 2014

Numerical Meth Engineering

View full text Add to dashboard Cite

We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. AbstractWe demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, i.e. weighted residual formulation, nodal basis functions on Gauss-Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocation and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established.

show abstract

An Ultra Weak Finite Element Method as an Alternative to a Monte Carlo Method for an Elasto-Plastic Problem with Noise

Abstract: International audienc

Cited by 16 publications

References 6 publications

Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise

Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise

A Feynman–Kac formula approach for computing expectations and threshold crossing probabilities of non-smooth stochastic dynamical systems

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Contact Info

Product

Resources

About

An Ultra Weak Finite Element Method as an Alternative to a Monte Carlo Method for an Elasto-Plastic Problem with Noise

Abstract: International audienc

Cited by 16 publications

References 6 publications

Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise

Penalization of a stochastic variational inequality modeling an elasto-plastic problem with noise

A Feynman–Kac formula approach for computing expectations and threshold crossing probabilities of non-smooth stochastic dynamical systems

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

Contact Info

Product

Resources

About

A collocated C⁰ finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics