An error estimator for real-time simulators based on model order reduction

Model order reduction (MOR) is one of the most appealing choices for real-time simulation of non-linear solids. In this work a method is presented in which real time performance is achieved by means of the o-line solution of a (high dimensional) parametric problem that provides a sort of response surface or computational vademecum. This solution is then evaluated in real-time at feedback rates compatible with haptic devices, for instance (i.e., more than 1kHz). This high dimensional problem can be solved without the limitations imposed by the curse of dimensionality by employing Proper Generalized Decomposition (PGD) methods. Essentially, PGD assumes a separated representation for the essential eld of the problem. Here, an error estimator is proposed for this type of solutions that takes into account the non-linear character of the studied problems. This error estimator allows to compute the necessary number of modes employed to obtain an approximation to the solution within a prescribed error tolerance in a given quantity of interest

Towards a pancreatic surgery simulator based on model order reduction

In this work a pancreatic surgery simulator is developed that provides the user with haptic feedback. The simulator is based on the use of model order reduction techniques, particularly Proper Generalized Decomposition methods. The just developed simulator presents some notable advancements with respect to existing works in the literature, such as the consideration of non-linear hyperelasticity for the constitutive modeling of soft tissues, an accurate description of contact between organs and momentum and energy conserving time integration schemes. Pancreas, liver, gall bladder, and duodenum are modeled in the simulator, thus providing with a very realistic and immersive perception to the user

A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems

An error estimator for real-time simulators based on model order reduction

Model order reduction (MOR) is one of the most appealing choices for real-time simulation of non-linear solids. In this work a method is presented in which real time performance is achieved by means of the o -line solution of a (high dimensional) parametric problem that provides a sort of response surface or computational vademecum. This solution is then evaluated in real-time at feedback rates compatible with haptic devices, for instance (i.e., more than 1kHz). This high dimensional problem can be solved without the limitations imposed by the curse of dimensionality by employing Proper Generalized Decomposition (PGD) methods. Essentially, PGD assumes a separated representation for the essential eld of the problem. Here, an error estimator is proposed for this type of solutions that takes into account the non-linear character of the studied problems. This error estimator allows to compute the necessary number of modes employed to obtain an approximation to the solution within a prescribed error tolerance in a given quantity of interest.Peer Reviewe

Simulation tools for biomechanical applications with PGD-based reduced order models

Cotutela Universitat Politècnica de Catalunya i Università degli Studi di PaviaNumerical simulation tools are generally used in all modern engineering fields, especially those having difficulties in performing large number of practical experiments, such as biomechanics. Among the computational methods, Finite Element (FE) is an essential tool. Nowadays, the fast-growing computational techniques, from the upgrading hardware to the emerging of novel algorithm, have already enabled extensive applications in biomechanics. For applications that require fast response and/or multiple queries, Reduced Order Modelling (ROM) methods have been developed based on existing methods such as FE, and have eventually enabled real-time numerical simulation for a large variety of engineering problems. In this thesis, several novel computational techniques are developed to explore the capability of Proper Generalised Decomposition (PGD), which is an important approach of ROM. To assess the usability of the PGD-based ROM for biomechanical applications, a real human femur bone is chosen to study its mechanical behaviour as an example. Standard image-based modelling procedure in biomechanics is performed to create an FE model which is then validated with in vitro experimental results. As a basis of this work, the medical image processing has to be performed, in order to generate an available FE model. This model is validated according to data collected from a previously performed \textit{in vitro} experimental test. The full procedure of image-based model generation and the validation of generated model is described in Chapter 2. As a major objective of this thesis, a non-intrusive scheme for the PGD framework is developed in Chapter 3. It is implemented using in-house developed Matlab (Mathworks, USA) code to conduct the PGD work flow, and calling Abaqus as an external solver for devised fictitious mechanical problems. The transformation of data from computed tomography (CT) image set to FE model including inhomogeneous material properties is subjected to some physical constraints, and when applying the load, there are also geometric constraints limiting the locations where load could be applied. These constraints will lead to a constrained parameter space, which possibly has difficulty to be separated in a Cartesian fashion. Therefore, a novel strategy to separate the parameters in a collective manner is proposed in Chapter 4. Chapter 5 details a comprehensive application in biomechanics, the methodologies proposed in Chapter 3 and 4 are applied on the practical model generated in Chapter 2. As a typical application of the PGD vademecum, a material property identification problem is discussed. Further PGD vademecum is generated using the identified material properties with variable loading locations, and with this vademecum, real-time mechanical response of the femur is available. In addition, for the purpose of extending the methodologies to orthotropic materials, which is commonly used in biomechanics, in Chapter 6 another linear elastic model is investigated with the non-intrusive PGD scheme. Nowadays, isogeometric analysis (IGA) is a very popular tool in computational mechanics. It is appealing to take advantage of non-uniform rational B-splines (NURBS) to discretise the model. For PGD, using B-splines for the discretisation of the parameter space could improve the quality of vademecum, especially for problems involving sensitivities with respect to the parameters during the online computations. It is important and necessary to extend the PGD framework to nonlinear solid mechanics, because most biological soft tissues have been observed nonlinear mechanical behaviours. Consequently, in Chapter 7 we have developed a PGD framework for the St.Venant-Kirchhoff constitutive model using the Picard linearisation which is consistent with the fixed-point iteration algorithm commonly used in PGD. In Chapter 8, conclusive remarks are addressed as well as forecasts of possible future works.Postprint (published version