Elías Cueto

Overview and recent advances in natural neighbour galerkin methods

SummaryIn this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods (also known as natural element methods, NEM), the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natural neighbour-based methods have certain unique features among the wide family of so-called meshless methods: a well-defined and robust approximation with no user-defined parameters on non-uniform grids, and the ability to exactly impose essential (Dirichlet) boundary conditions are particularly noteworthy.A comprehensive review of the method is conducted, including a description of the Sibson and the Laplace interpolants in two- and three-dimensions. Application of the NEM to linear and non-linear problems in solid as well as fluid mechanics is studied. Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark computations that demonstrate the accuracy and the key advantages of this numerical method.

On the "A priori" model reduction : overview and recent developments.

SummaryKarhunen-Loève expansion and snapshot POD are based on principal component analysis of series of data. They provide basis vectors of the subspace spanned by the data. All the data must be taken into account to find the basis vectors. These methods are not convenient for any improvement of the basis vectors when new data are added into the data base. We consider the data as a state evolution and we propose an incremental algorithm to build basis functions for the decomposition of this state evolution. The proposed algorithm is based on the APHR method (A Priori Hyper-Reduction method). This is an adaptive strategy to build reduced order model when the state evolution is implicitely defined by non-linear governing equations. In case of known state evolutions the APHR method is an incremental Karhunen-Loève decomposition. This approach is very convenient to expand the subspace spanned by the basis functions. In the first part of the present paper the main concepts related to the “a priori” model reduction technique are revisited, as a previous task to its application in the cases considered in the next sections.Some engineering problems are defined in domains that evolve in time. When this evolution is large the present and the reference configurations differ significantly. Thus, when the problem is formulated in the total Lagrangian framework frequent remeshing is required to avoid too large distortions of the finite element mesh. Other possibility for describing these models lies in the use of an updated formulation in which the mesh is conformed to each intermediate configuration. When the finite element method is used, then frequent remeshing must be carried out to perform an optimal meshing at each intermediate configuration. However, when the natural element method, a novel meshless technique, is considered, whose accuracy does not depend significantly on the relative position of the nodes, then large simulations can be performed without any remeshing stage, being the nodal position at each intermediate configuration defined by the transport of the nodes by the material velocity or the advection terms. Thus, we analyze the extension of the “a priori” model reduc tion, based on the use in tandem of the Karhunen-Loève decomposition (that extracts significant information) and an approximation basis enrichment based on the use of the Krylovs subspaces, previously proposed in the framework of fixed mesh simulation, to problems defined in domains evolving in time.Finally, for illustrating the technique capabilities, the “a priori” model reduction will be applied for solving the kinetic theory model which governs the orientation of the fibers immersed in a Newtonian flow.

Real-time deformable models of non-linear tissues by model reduction techniques

In this paper we introduce a new technique for the real-time simulation of non-linear tissue behavior based on a model reduction technique known as proper orthogonal (POD) or Karhunen-Loève decompositions. The technique is based upon the construction of a complete model (using finite element modelling or other numerical technique, for instance, but possibly from experimental data) and the extraction and storage of the relevant information in order to construct a model with very few degrees of freedom, but that takes into account the highly non-linear response of most living tissues. We present its application to the simulation of palpation a human cornea and study the limitations and future needs of the proposed technique.

Accounting for large deformations in real-time simulations of soft tissues based on reduced-order models

Model reduction techniques have shown to constitute a valuable tool for real-time simulation in surgical environments and other fields. However, some limitations, imposed by real-time constraints, have not yet been overcome. One of such limitations is the severe limitation in time (established in 500Hz of frequency for the resolution) that precludes the employ of Newton-like schemes for solving non-linear models as the ones usually employed for modeling biological tissues. In this work we present a technique able to deal with geometrically non-linear models, based on the employ of model reduction techniques, together with an efficient non-linear solver. Examples of the performance of the technique over some examples will be given.

Real Time Simulation of Biological Soft Tissues: A PGD Approach

We introduce here a novel approach for the numerical simulation of nonlinear, hyperelastic soft tissues at kilohertz feedback rates necessary for haptic rendering. This approach is based upon the use of proper generalized decomposition techniques, a generalization of PODs. Proper generalized decomposition techniques can be considered as a means of a priori model order reduction and provides a physics-based meta-model without the need for prior computer experiments. The suggested strategy is thus composed of an offline phase, in which a general meta-model is computed, and an online evaluation phase in which the results are obtained at real time. Results are provided that show the potential of the proposed technique, together with some benchmark test that shows the accuracy of the method.

Original article: Proper Generalized Decomposition based dynamic data driven inverse identification

Dynamic Data-Driven Application Systems-DDDAS-appear as a new paradigm in the field of applied sciences and engineering, and in particular in Simulation-based Engineering Sciences. By DDDAS we mean a set of techniques that allow the linkage of simulation tools with measurement devices for real-time control of systems and processes. One essential feature of DDDAS is the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability of an application to dynamically control the measurement process. DDDAS need accurate and fast simulation tools using if possible off-line computations to limit as much as possible the on-line computations. With this aim, efficient solvers can be constructed by introducing all the sources of variability as extra-coordinates in order to solve the model off-line only once. This way, its most general solution is obtained and therefore it can be then considered in on-line purposes. So to speak, we introduce a physics-based meta-modeling technique without the need for prior computer experiments. However, such models, that must be solved off-line, are defined in highly multidimensional spaces suffering the so-called curse of dimensionality. We proposed recently a technique, the Proper Generalized Decomposition-PGD-able to circumvent the redoubtable curse of dimensionality. The marriage of DDDAS concepts and tools and PGD off-line computations could open unimaginable possibilities in the field of dynamic data-driven application systems. In this work we explore some possibilities in the context of on-line parameter estimation.

Real‐time simulation of surgery by reduced‐order modeling and X‐FEM techniques

This paper describes a novel approach for the simulation of surgery by a combined technique of model order reduction and extended finite element method (X-FEM) methods. Whereas model order reduction techniques employ globally supported (Ritz) shape functions, a combination with X-FEM methods on a locally superimposed patch is developed for cutting simulation without remeshing. This enables to obtain models with very few degrees of freedom that run under real-time constrains even for highly non-linear tissue constitutive equations. To show the performance of the technique, we studied an application to refractive surgery in the cornea.

Natural element meshless simulation of flows involving short fiber suspensions

Abstract Numerical modeling of non-Newtonian flows typically involves the coupling between the equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics (coupled problem). Some industrial flows involve moving or free boundaries (injection, extrusion, …). Lagrangian descriptions allow an accurate description of the flow front tracking as well as an accurate integration of transport equations along the flow trajectories. However, Lagrangian techniques in the context of finite elements have the important drawback of requiring frequent remeshing in order to avoid large elements distortions. The natural element method (NEM) has the capabilities of Lagrangian models to describe the flow front tracking as well as to treat the convection terms related to the fiber orientation equation without the mesh quality requirement characteristics of the standard finite elements method.

Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions.

The numerical solution of the chemical master equation (CME) governing gene regulatory networks and cell signaling processes remains a challenging task owing to its complexity, exponentially growing with the number of species involved. Although most of the existing techniques rely on the use of Monte Carlo-like techniques, we present here a new technique based on the approximation of the unknown variable (the probability of having a particular chemical state) in terms of a finite sum of separable functions. In this framework, the complexity of the CME grows only linearly with the number of state space dimensions. This technique generalizes the so-called Hartree approximation, by using terms as needed in the finite sums decomposition for ensuring convergence. But noteworthy, the ease of the approximation allows for an easy treatment of unknown parameters (as is frequently the case when modeling gene regulatory networks, for instance). These unknown parameters can be considered as new space dimensions. In this way, the proposed method provides solutions for any value of the unknown parameters (within some interval of arbitrary size) in one execution of the program.

Real-time monitoring of thermal processes by reduced-order modeling

1 Institut de Recherche en Génie Civil et Mécanique (GeM UMR CNRS 6183), Ecole Centrale de Nantes. 1 rue de la Noë, BP 92101, F-44321 Nantes cedex 3, France. e-mail: {jose.aguado-lopez,francisco.chinesta}@ec-nantes.fr, web http://rom.ec-nantes.fr 2Laboratori de Calcul Numeric (LaCaN). Departament de Matematica Aplicada III E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universitat Politecnica de Catalunya, BarcelonaTech, 08034 Barcelona, Spain. e-mail: antonio.huerta@upc.edu, web http://www.lacan.upc.edu 3Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Swansea SA2 8PP, UK. 4Aragon Institute of Engineering Research (I3A), Universidad de Zaragoza, Maria de Luna 3, E-50018 Zaragoza, Spain. e-mail: ecueto@unizar.es, web http://amb.unizar.es