Michael Ortiz “Model-Free Data-Driven Computing”

It was a pleasure to welcome Michael Ortiz from the California Institute of Technology to give a seminar on the topic of “Model-Free Data-Driven Computing” at the University of Luxembourg. You can watch the entire seminar below.

Abstract:

We develop a new computing paradigm, which we refer to as Data-Driven Computing, according to which calculations are carried out directly from experimental material data and pertinent kinematic constraints and conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We demonstrate the data-driven paradigm and investigate the performance of data-driven solvers by means of several examples of application, including statics and dynamics of nonlinear three-dimensional trusses, and linear and nonlinear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment, including noisy material data sets containing outliers. The variational structure of the data-driven problem also renders it amenable to analysis. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within the Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to effective material data sets that are not graphs. I will finish my presentation with highlights on work in progress, including closed-loop Data-Driven analysis and experiments, Data-Driven molecular dynamics, Data-Driven inelasticity and publicly-editable material data repositories and data management from a Data-Driven perspective.

Acoustic/Electromagnetic scattering

Acoustic/Electromagnetic scattering
WMtot

Electromagnetic field generated inside human brain

The purpose of this project is to reduce the errors associated with numerical solution of wave propagation problems and their computational costs. In high-frequency regimes, the solution of conventional FEM sufferers from pollution error which is due to dispersion and can be visualized as a phase shift of the numerical solution. To maintain a desired level of pollution error in conventional FEM, it is necessary to increase discretization density faster than the wave number which rapidly increase the computational cost. On the other hand, the geometrical accuracy of the scattering surfaces play an important role in accuracy of the solution. The possibility of representing man-made objects exactly in IGA even with very coarse meshes and the convenience of its refinement makes it a desirable platform to perform scattering analysis. The convergence graphs obtained for scattering problems performed in IGA platform approves the anticipated properties and makes it possible to increase the solution accuracy even for very high-frequency analysis. [http://mechanical907.rssing.com/browser.php?indx=25030677&last=1&item=1], [http://hdl.handle.net/10993/28982].

Sphere_real

Exterior acoustic scattering

p10_IGA

Exterior acoustic scattering problem – The evolution of L2 error with discretization density in IGA. High-order IGA yields low error with very coarse meshes.

Accelerating Monte Carlo estimation with derivatives of high-level finite element models. CMAME paper submitted with Jack and Paul. 

http://hdl.handle.net/10993/28618 Link

In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a one- dimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney-Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.