Numerical Computation, Data Analysis and Software in Mathematics and Engineering

The present book contains 14 articles that were accepted for publication in the Special Issue “Numerical Computation, Data Analysis and Software in Mathematics and Engineering” of the MDPI journal Mathematics. The topics of these articles include the aspects of the meshless method, numerical simulation, mathematical models, deep learning and data analysis. Meshless methods, such as the improved element-free Galerkin method, the dimension-splitting, interpolating, moving, least-squares method, the dimension-splitting, generalized, interpolating, element-free Galerkin method and the improved interpolating, complex variable, element-free Galerkin method, are presented. Some complicated problems, such as tge cold roll-forming process, ceramsite compound insulation block, crack propagation and heavy-haul railway tunnel with defects, are numerically analyzed. Mathematical models, such as the lattice hydrodynamic model, extended car-following model and smart helmet-based PLS-BPNN error compensation model, are proposed. The use of the deep learning approach to predict the mechanical properties of single-network hydrogel is presented, and data analysis for land leasing is discussed. This book will be interesting and useful for those working in the meshless method, numerical simulation, mathematical model, deep learning and data analysis fields

Modeling Strong Discontinuities Using Generalized Finite Element Method (GFEM)

Computational modeling of strong discontinuities is a challenging a task. Stress analysis and shape sensitivity can be done with any simulation method such as finite element method, finite difference method, Rayleigh-Ritz method, weighted residual method or least square method. All these numerical analysis methods are computationally expensive and uneconomical. Mesh dependency with respect to fracture boundary is a major problem in these methods. Furthermore accuracy of the methods depends on the size of the discretized element

Adaptive smoothed stable extended finite element method for weak discontinuities for finite elasticity

In this paper, we propose a smoothed stable extended finite element method (S2XFEM) by combining the strain smoothing with the stable extended finite element method (SXFEM) to efficiently treat inclusions and/or voids in hyperelastic matrix materials. The interface geometries are implicitly represented through level sets and a geometry based error indicator is used to resolve the geometry. For the unknown fields, the mesh is refined based on a recovery based error indicator combined with a quadtree decomposition guarantee the method’s accuracy with respect to the computational costs. Elements with hanging nodes (due to the quadtree meshes) are treated as polygonal elements with mean value coordinates as the basis functions. The accuracy and the convergence properties are compared to similar approaches for several numerical examples. The examples indicate that S2XFEM is computationally the most efficient without compromising the accuracy

Multi-resolution methods for high fidelity modeling and control allocation in large-scale dynamical systems

This dissertation introduces novel methods for solving highly challenging model- ing and control problems, motivated by advanced aerospace systems. Adaptable, ro- bust and computationally effcient, multi-resolution approximation algorithms based on Radial Basis Function Network and Global-Local Orthogonal Mapping approaches are developed to address various problems associated with the design of large scale dynamical systems. The main feature of the Radial Basis Function Network approach is the unique direction dependent scaling and rotation of the radial basis function via a novel Directed Connectivity Graph approach. The learning of shaping and rota- tion parameters for the Radial Basis Functions led to a broadly useful approximation approach that leads to global approximations capable of good local approximation for many moderate dimensioned applications. However, even with these refinements, many applications with many high frequency local input/output variations and a high dimensional input space remain a challenge and motivate us to investigate an entirely new approach. The Global-Local Orthogonal Mapping method is based upon a novel averaging process that allows construction of a piecewise continuous global family of local least-squares approximations, while retaining the freedom to vary in a general way the resolution (e.g., degrees of freedom) of the local approximations. These approximation methodologies are compatible with a wide variety of disciplines such as continuous function approximation, dynamic system modeling, nonlinear sig-nal processing and time series prediction. Further, related methods are developed for the modeling of dynamical systems nominally described by nonlinear differential equations and to solve for static and dynamic response of Distributed Parameter Sys- tems in an effcient manner. Finally, a hierarchical control allocation algorithm is presented to solve the control allocation problem for highly over-actuated systems that might arise with the development of embedded systems. The control allocation algorithm makes use of the concept of distribution functions to keep in check the "curse of dimensionality". The studies in the dissertation focus on demonstrating, through analysis, simulation, and design, the applicability and feasibility of these ap- proximation algorithms to a variety of examples. The results from these studies are of direct utility in addressing the "curse of dimensionality" and frequent redundancy of neural network approximation

Spectral, Combinatorial, and Probabilistic Methods in Analyzing and Visualizing Vector Fields and Their Associated Flows

In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\'' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration

Numerical techniques for near-field acoustic holography

Near-field acoustic holography (NAH) has proved to be an enormously successful sound source identification technique, which is widely used in field of acoustics and noise control. The work presented in this thesis aims at developing a novel method for modelling near-field acoustic holography, where the particle velocity on the surface of a vibrating structure is recovered from measurements taken in the exterior field. The model developed will form a powerful predictive tool for identifying sources of acoustic radiation for applications in mechanical and audio engineering. Inspired by advances in the solution of ill-posed problems in imaging science using the so-called compressed sensing, we seek to develop new compressed or sparse reconstruction methods for the NAH problem. A sparse superposition method will be developed and implemented based on the inverse method of superposition (MoS), or equivalent source method as it is often known. The method should be able to reconstruct the normal velocity of a vibrating object using a very small number of charge points. Two primary reasons this is beneficial are; one can potentially reduce the amount of measured data required, and one could also detect sources of noise when small clusters of charge points are identified. The sparse inverse MoS will then be applied to reconstruct the surface velocity of a loudspeaker from measurements of the sound pressure field taken in a semi-anechoic chamber. The development of the new sparse inverse MoS and its experimental verification form the primary contributions to knowledge of this thesis

Gaussian Process Modeling for Upsampling Algorithms With Applications in Computer Vision and Computational Fluid Dynamics

Across a variety of fields, interpolation algorithms have been used to upsample lowresolution or coarse data fields. In this work, novel Gaussian Process based methodsare employed to solve a variety of upsampling problems. Specifically threeapplications are explored: coarse data prolongation in Adaptive Mesh Refinement(AMR) in the field of Computational Fluid Dynamics, accurate document imageupsampling to enhance Optical Character Recognition (OCR) accuracy, and fastand accurate Single Image Super Resolution (SISR). For AMR, a new, efficient,and “3rd order accurate” algorithm called GP-AMR is presented. Next, a novel,non-zero mean, windowed GP model is generated to upsample low resolution documentimages to generate a higher OCR accuracy, when compared to the industrystandard. Finally, a hybrid GP convolutional neural network algorithm is used togenerate a computationally efficient and high quality SISR model

Numerical investigation of bone adaptation to exercise and fracture in Thoroughbred racehorses

Third metacarpal bone (MC3) fracture has a massive welfare and economic impact on horse racing, representing 45% of all fatal lower limb fractures, which in themselves represent more than 80% of reasons for death or euthanasia on the UK racecourses. Most of these fractures occur due to the accumulation of tissue fatigue as a result of repetitive loading rather than a specific traumatic event. Despite considerable research in the field, including applying various diagnostic methods, it still remains a challenge to accurately predict the fracture risk and prevent this type of injury. The objective of this thesis is to develop computational tools to quantify bone adaptation and resistance to fracture, thereby providing the basis for a viable and robust solution. Recent advances in subject-specific finite element model generation, for example computed tomography imaging and efficient segmentation algorithms, have significantly improved the accuracy of finite element modelling. Numerical analysis techniques are widely used to enhance understanding of fracture in bones and provide better insight into relationships between load transfer and bone morphology. This thesis proposes a finite element based framework allowing for integrated simulation of bone remodelling under specific loading conditions, followed by the evaluation of its fracture resistance. Accurate representation of bone geometry and heterogeneous material properties are obtained from calibrated computed tomography scans.The material mapping between CT-scan data and discretised geometries for the finite element method is carried out by using Moving Least Squares approximation and L2-projection. Thus is then used for numerical investigations and assessment of density gradients at the common site of fracture. Bone is able to adapt its density to changes in external conditions. This property is one of the most important mechanisms for the development of resistance to fracture. Therefore, a finite element approach for simulating adaptive bone changes (also called bone remodelling) is proposed. The implemented method is based on a phenomenological model of the macroscopic behaviour of bone based on the thermodynamics of open systems. Numerical results showed that the proposed technique has the potential to accurately simulate the long-term bone response to specified training conditions and also improve possible treatment options for bone implants. Assessment of the fracture risk was conducted with crack propagation analysis. The potential of two different approaches was investigated: smeared phase-field and discrete configurational mechanics approach. The popular phase-field method represents a crack by a smooth damage variable leading to a phase-field approximation of the variational formulation for brittle fracture. A robust solution scheme was implemented using a monolithic solution scheme with arc-length control. In the configurational mechanics approach, the driving forces, and fracture energy release rate, are expressed in terms of nodal quantities, enabling a fully implicit formulation for modelling the evolving crack front. The approach was extended for the first time to capture the influence of heterogeneous density distribution. The outcomes of this study showed that discrete and smeared crack approximations are capable of predicting crack paths in three-dimensional heterogeneous bodies with comparable results. However, due to the necessity of using significantly finer meshes, phase-field was found to be less numerically efficient. Finally, the current state of the framework's development was assessed using numerical simulations for bone adaptation and subsequent fracture propagation, including analysis of an equine metacarpal bone. Numerical convergence was demonstrated for all examples, and the use of singularity elements proved to further improve the rate of convergence. It was shown that bone adaptation history and bone density distribution influence both fracture resistance and the resulting crack path. The promising results of this study offer a~novel framework to simulate changes in the bone structure in response to exercise and quantify the likelihood of a fracture