A Meshfree Generalized Finite Difference Method for Surface PDEs

In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinous surface properties (with large jumps) are also introduced, and a few practical applications are presented

From medical images to individualized cardiac mechanics: A Physiome approach

Cardiac mechanics is a branch of science that deals with forces, kinematics, and material properties of the heart, which is valuable for clinical applications and physiological studies. Although anatomical and biomechanical experiments are necessary to provide the fundamental knowledge of cardiac mechanics, the invasive nature of the procedures limits their further applicability. In consequence, noninvasive alternatives are required, and cardiac images provide an excellent source of subject-specific and in vivo information. Noninvasive and individualized cardiac mechanical studies can be achieved through coupling general physiological models derived from invasive experiments with subject-specific information extracted from medical images. Nevertheless, as data extracted from images are gross, sparse, or noisy, and do not directly provide the information of interest in general, the couplings between models and measurements are complicated inverse problems with numerous issues need to be carefully considered. The goal of this research is to develop a noninvasive framework for studying individualized cardiac mechanics through systematic coupling between cardiac physiological models and medical images according to their respective merits. More specifically, nonlinear state-space filtering frameworks for recovering individualized cardiac deformation and local material parameters of realistic nonlinear constitutive laws have been proposed. To ensure the physiological meaningfulness, clinical relevance, and computational feasibility of the frameworks, five key issues have to be properly addressed, including the cardiac physiological model, the heart representation in the computational environment, the information extraction from cardiac images, the coupling between models and image information, and also the computational complexity. For the cardiac physiological model, a cardiac physiome model tailored for cardiac image analysis has been proposed to provide a macroscopic physiological foundation for the study. For the heart representation, a meshfree method has been adopted to facilitate implementations and spatial accuracy refinements. For the information extraction from cardiac images, a registration method based on free-form deformation has been adopted for robust motion tracking. For the coupling between models and images, state-space filtering has been applied to systematically couple the models with the measurements. For the computational complexity, a mode superposition approach has been adopted to project the system into an equivalent mathematical space with much fewer dimensions for computationally feasible filtering. Experiments were performed on both synthetic and clinical data to verify the proposed frameworks

Investigation of the use of meshfree methods for haptic thermal management of design and simulation of MEMS

This thesis presents a novel approach of using haptic sensing technology combined with virtual environment (VE) for the thermal management of Micro-Electro-Mechanical-Systems (MEMS) design. The goal is to reduce the development cycle by avoiding the costly iterative prototyping procedure. In this regard, we use haptic feedback with virtua lprototyping along with an immersing environment. We also aim to improve the productivity and capability of the designer to better grasp the phenomena operating at the micro-scale level, as well as to augment computational steering through haptic channels. To validate the concept of haptic thermal management, we have implemented a demonstrator with a user friendly interface which allows to intuitively "feel" the temperature field through our concept of haptic texturing. The temperature field in a simple MEMS component is modeled using finite element methods (FEM) or finite difference method (FDM) and the user is able to feel thermal expansion using a combination of different haptic feedback. In haptic application, the force rendering loop needs to be updated at a frequency of 1Khz in order to maintain continuity in the user perception. When using FEM or FDM for our three-dimensional model, the computational cost increases rapidly as the mesh size is reduced to ensure accuracy. Hence, it constrains the complexity of the physical model to approximate temperature or stress field solution. It would also be difficult to generate or refine the mesh in real time for CAD process. In order to circumvent the limitations due to the use of conventional mesh-based techniques and to avoid the bothersome task of generating and refining the mesh, we investigate the potential of meshfree methods in the context of our haptic application. We review and compare the different meshfree formulations against FEM mesh based technique. We have implemented the different methods for benchmarking thermal conduction and elastic problems. The main work of this thesis is to determine the relevance of the meshfree option in terms of flexibility of design and computational charge for haptic physical model

Concurrently coupled solid shell-based adaptive multiscale method for fracture

Artículo Open Access en el sitio web del editor. Pago por publicar en abierto.A solid shell-based adaptive atomistic–continuum numerical method is herein proposed to simulate complex crack growth patterns in thin-walled structures. A hybrid solid shell formulation relying on the combined use of the enhanced assumed strain (EAS) and the assumed natural strain (ANS) methods has been considered to efficiently model the material in thin structures at the continuum level. The phantom node method (PNM) is employed to model the discontinuities in the bulk. The discontinuous solid shell element is then concurrently coupled with a molecular statics model placed around the crack tip. The coupling between the coarse scale and the fine scale is realized through the use of ghost atoms, whose positions are interpolated from the coarse scale solution and enforced as boundary conditions to the fine scale model. In the proposed numerical scheme, the fine scale region is adaptively enlarged as the crack propagates and the region behind the crack tip is adaptively coarsened in order to reduce the computation costs. An energy criterion is used to detect the crack tip location. All the atomistic simulations are carried out using the LAMMPS software. A computational framework has been developed in MATLAB to trigger LAMMPS through system command. This allows a two way interaction between the coarse and fine scales in MATLAB platform, where the boundary conditions to the fine region are extracted from the coarse scale, and the crack tip location from the atomistic model is transferred back to the continuum scale. The developed framework has been applied to study crack growth in the energy minimization problems. Inspired by the influence of fracture on current–voltage characteristics of thin Silicon photovoltaic cells, the cubic diamond lattice structure of Silicon is used to model the material in the fine scale region, whilst the Tersoff potential function is employed to model the atom–atom interactions. The versatility and robustness of the proposed methodology is demonstrated by means of several fracture applications.Unión Europea ERC 306622Ministerio de Economía y Competitividad DPI2012-37187, MAT2015-71036-P y MAT2015-71309-PJunta de Andalucía P11-TEP-7093 y P12-TEP -105

Personalized noninvasive imaging of volumetric cardiac electrophysiology

Three-dimensionally distributed electrical functioning is the trigger of mechanical contraction of the heart. Disturbance of this electrical flow is known to predispose to mechanical catastrophe but, due to its amenability to certain intervention techniques, a detailed understanding of subject-specific cardiac electrophysiological conditions is of great medical interest. In current clinical practice, body surface potential recording is the standard tool for diagnosing cardiac electrical dysfunctions. However, successful treatments normally require invasive catheter mapping for a more detailed observation of these dysfunctions. In this dissertation, we take a system approach to pursue personalized noninvasive imaging of volumetric cardiac electrophysiology. Under the guidance of existing scientific knowledge of the cardiac electrophysiological system, we extract the subject specific cardiac electrical information from noninvasive body surface potential mapping and tomographic imaging data of individual subjects. In this way, a priori knowledge of system physiology leads the physiologically meaningful interpretation of personal data; at the same time, subject-specific information contained in the data identifies parameters in individual systems that differ from prior knowledge. Based on this perspective, we develop a physiological model-constrained statistical framework for the quantitative reconstruction of the electrical dynamics and inherent electrophysiological property of each individual cardiac system. To accomplish this, we first develop a coupled meshfree-BE (boundary element) modeling approach to represent existing physiological knowledge of the cardiac electrophysiological system on personalized heart-torso structures. Through a state space system approach and sequential data assimilation techniques, we then develop statistical model-data coupling algorithms for quantitative reconstruction of volumetric transmembrane potential dynamics and tissue property of 3D myocardium from body surface potential recoding of individual subjects. We also introduce a data integration component to build personalized cardiac electrophysiology by fusing tomographic image and BSP sequence of the same subject. In addition, we develop a computational reduction strategy that improves the efficiency and stability of the framework. Phantom experiments and real-data human studies are performed for validating each of the framework’s major components. These experiments demonstrate the potential of our framework in providing quantitative understanding of volumetric cardiac electrophysiology for individual subjects and in identifying latent threats in individual’s heart. This may aid in personalized diagnose, treatment planning, and fundamentally, prevention of fatal cardiac arrhythmia

Meshfree modelling of metal cutting using phenomenological and data-driven material models

Numerical modelling of chip formation is important for a better understanding thus for improvement of the high speed metal cutting process. The objective of this work is to develop a meshfree numerical simulation framework for the chip formation process, where both the phenomenological material model and the data-driven material model can be applied. Firstly, the phenomenological model is applied to capture the serrated chip formation, where a recently developed Galerkin type meshfree scheme, the stabilized Optimal Transportation Meshfree (OTM) method, is applied as a numerical solution method. This enables the modelling of material separation and serrated morphology generation of the cutting process in a more realistic and convenient way. The shear band formation is described by the thermal softening term in the Johnson-Cook plastic flow stress model. Using this model, it can be demonstrated that thermal softening is the main cause of the shear band formation. Additionally, it can be seen that the Johnson-Cook fracture model shows limitations in capturing the fracture on the chip upper surface. Thus, a supplementary condition for the stress triaxiality is applied. This condition allows more accurate measurements of the chip size, i.e. chip spacing, peak and valley. These improvements are demonstrated by comparing the calculated chip morphology, cutting force and chip formation process with experimental results. Subsequently, the data-driven material model is developed to replace the classical material model, where the Machine Learning (ML) based model is first trained offline to fit an observed material behaviour and later be used in online applications. However, learning and predicting history dependent material models such as plasticity is still challenging. In this work, a ML based material modelling framework is proposed for both elasticity and plasticity, in which the ML based hyperelasticity model is directly developed with the Feed forward Neural Network (FNN), whereas the ML based plasticity model is developed by using two approaches including Recurrent Neural Network (RNN) and a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN). In the latter case, the accumulated absolute strain is proposed to distinguish the loading history. Additionally, the strain-stress sequence data for the plasticity model is collected from different loading paths based on the concept of sequence for plasticity. By means of the POD, the multi-dimensional stress sequence is decoupled as independent one dimensional coefficient sequences. To apply the ML based material model in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiation tool AceGen. The effectiveness and generalisation of the presented models are investigated by a series of numerical examples using both 2D and 3D finite element analysis. Finally, the ML based material model is applied to the metal cutting simulation

An adaptive continuum/discrete crack approach for meshfree particle methods

A coupled continuum/discrete crack model for strain softening materials is implemented in a meshfree particle code. A coupled damage plasticity constitutive law is applied until a certain strain based threshold value - this is at the maximum tensile stress of the equivalent uniaxial stress strain curve - is reached. At this point a discrete crack is introduced and described as an internal boundary with a traction crack opening relation. Within the frame-work of meshfree particle methods it is possible to model the transition from the continuum to the discrete crack since boundaries and particles can easily be added and removed. The EFG method and an explicit time integration scheme is used. The integrals are evaluated by nodal integration, an integration with stress points and also a full Gauss quadrature. Some results are compared to experimental data and show good agreement. Additional comparisons are made to a pure continuum constitutive law

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Bayesian Inference with Combined Dynamic and Sparsity Models: Application in 3D Electrophysiological Imaging

Data-driven inference is widely encountered in various scientific domains to convert the observed measurements into information that cannot be directly observed about a system. Despite the quickly-developing sensor and imaging technologies, in many domains, data collection remains an expensive endeavor due to financial and physical constraints. To overcome the limits in data and to reduce the demand on expensive data collection, it is important to incorporate prior information in order to place the data-driven inference in a domain-relevant context and to improve its accuracy. Two sources of assumptions have been used successfully in many inverse problem applications. One is the temporal dynamics of the system (dynamic structure). The other is the low-dimensional structure of a system (sparsity structure). In existing work, these two structures have often been explored separately, while in most high-dimensional dynamic system they are commonly co-existing and contain complementary information. In this work, our main focus is to build a robustness inference framework to combine dynamic and sparsity constraints. The driving application in this work is a biomedical inverse problem of electrophysiological (EP) imaging, which noninvasively and quantitatively reconstruct transmural action potentials from body-surface voltage data with the goal to improve cardiac disease prevention, diagnosis, and treatment. The general framework can be extended to a variety of applications that deal with the inference of high-dimensional dynamic systems