Abstract

Despite the impressive progresses attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to high-dimensional models, on which mesh-based approaches fail due to the exponential increase of degrees of freedom. Other challenging scenarios concern problems requiring many direct solutions (optimization, inverse identification, uncertainty quantification …) or those needing very fast solutions (real time simulation, simulation based control …).

We are developing a novel technique, called Proper Generalized Decomposition (PGD) based on the assumption of a separated form of the unknown fields that has demonstrated its capabilities in dealing with high-dimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for addressing standard problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework opening new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of “Computational Vademecum”. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the “vademecum”, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries. 

Abstract

Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in material properties are frequently modeled through random fields, which typically induces the need to solve finite element problems for a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at an affordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittle materials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition (PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution for the physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a postprocessing operation on the explicit solution provided by PGD.We first validate the framework using a global energy approach on a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced order modeling approach to a stochastic fracture propagation problem.

Abstract

In this paper we present the basics of a novel methodology for the development of simulation-based and augmented learning tools in the context of applied science and engineering. It is based on the extensive use of model order reduction, and particularly, of the so-called Proper Generalized Decomposition (PGD) method. This method provides a sort of meta-modeling tool without the need for prior computer experiments that allows the user to obtain real-time response in the solution of complex engineering or physical problems. This real-time capability also allows for its implementation in deployed, touch-screen, handheld devices or even to be immersed into electronic textbooks. We explore here the basics of the proposed methodology and give examples on a few challenging applications never until now explored, up to our knowledge.

Abstract

It is well known that model order reduction techniques that project the solution of the problem at hand onto alow-dimensional subspace present difficulties when this solution lies on a non-linear manifold. To overcomethese difficulties—notably, an undesirable augment in the number of requiredmodesin the solution—severalsolutions have been suggested. Among them we can cite the use of non-linear dimensionality reductiontechniques or, alternatively, the employ of local linear reduced order approaches. These last approachesusually present the difficulty of ensuring continuity between these local models. Here, a new method ispresented that ensures this continuity by resorting to the paradigm of the partition of unity, while employing Proper Generalized Decompositions at each local patch.

Abstract

Many harbor applications are based on the solution of linear elliptic agitation problems for many spectral conditions. One of the main goals consists in computing the linear combination of numerous simulations of the harbor agitation problem, using monochromatic waves of different spectral components (i.e. frequency and incoming wave direction). In practice, the standard strategy selects the number of wave components according to a prescribed discretization of the 2D input spectra. The main issue relies on some quantities of interest that are very sensitive to the level of refinement of the spectra, such as the significant wave height at every mesh point or the identification of resonance modes induced by long wave scattering. In many cases, achieving enough quality in these quantities may impose numerous simulations and, consequently, non-practical computer costs. This can drastically limit the final accuracy of results. To overcome this situation, here a new strategy is proposed to efficiently solve a large number of harbor agitation problems derived from dense discretizations of the 2D input spectra. The strategy is based on the combination of two different numerical approaches. Firstly, each required monochromatic simulation is solved via high order NURBS (non-uniform rational B-splines) enhanced finite elements (NEFEM). More precisely, NEFEM captures the exact harbor geometry using large mesh elements that produce accurate solutions and significant savings on the system size, particularly in long wave cases. Secondly, a model order reduction technique is used to approximate the original elliptic harbor model by a so-called surrogate model. The main advantage is that, once the surrogate model is constructed, it can be rapidly evaluated to provide simulations for any value of the spectral components within a range of interest, and without the need of solving any new harbor agitation problem (as the standard strategy does). Thus, this enables the possibility of using any desired discretization of the 2D input spectra with no additional computer cost. The construction of the surrogate model is performed using the proper generalized decomposition method with a novel incremental computation along the frequency dimension. The proposed strategy is discussed, and its superior performance with respect to standard strategies is demonstrated, on two harbor agitation examples with several applications.

Abstract

The Parameterized Background Data-Weak (PBDW) method is a non-intrusive, reduced basis, real-time and in-situ data assimilation framework for physical systems modeled by parametrized Partial Differential Equations for steady-state problems. The key idea of the formulation is to seek an approximation to the true field employing projection-by-data, with a first contribution from a deduced background estimate from reduced modeling and a second contribution from an update state informed by the experimental observations (correction of model bias). The present study aims at extending the PBDW formulation for time-dependent problems and proposes a sequential version to deal with sampled data. The work focuses on a time integration in the reduced order model, a data-driven empirical enrichment of the model and a state prediction for future time steps.