Towards New High-Order Operator Splitting Time-Integration Methods

Operator splitting (OS) methods represent a powerful strategy to solve an extensive range of mathematical models in the form of differential equations. They have a long and celebrated history, having been successfully used for well over half a century to provide efficient numerical solutions to challenging problems. In fact, OS methods are often the only viable way to solve many problems in practice. The simplest, and perhaps, most well-known OS methods are Lie--Trotter--Godunov and the Strang--Marchuk methods. They compute a numerical solution that is first-, and second-order accurate in time, respectively. OS methods can be derived by imposing order conditions using the Campbell--Baker--Hausdorff formula. It follows that, by setting the appropriate order conditions, it is possible to derive OS methods of any desired order. An important observation regarding OS methods with order higher than two is that, according to the Sheng--Suzuki theorem, at least one of their defining coefficients must be negative. Therefore, the time integration with OS methods of order higher than two has not been considered suitable to solve deterministic parabolic problems, because the necessary backward time integration would cause instabilities. Throughout this thesis, we focus our attention on high-order (i.e., order higher than two) OS methods. We successfully assess the convergence properties of some higher-order OS methods on diffusion-reaction problems describing cardiac electrophysiology and on an advection-diffusion-reaction problem describing chemical combustion. Furthermore, we compare the efficiency performance of higher-order methods to second-order methods. For all the cases considered, we confirm an improved efficiency performance compared to methods of lower order. Next, we observe how, when using OS and Runge--Kutta type methods to advance the time integration, we can construct a unique extended Butcher tableau with a similar structure to the ones describing Generalized Additive Runge--Kutta (GARK) methods. We define a combination of methods to be OS-GARK methods. We apply linear stability analysis to OS-GARK methods; this allows us to conveniently analyze the stability properties of any combination of OS and Runge--Kutta methods. Doing so, we are able to perform an eigenvalue analysis to understand and improve numerically unstable solutions

Dynamics meets Morphology: towards Dymorph Computation

In this dissertation, approaches are presented for both technically using and investigating biological principles with oscillators in the context of electrical engineering, in particular neuromorphic engineering. Thereby, dynamics as well as morphology as important neuronal principles were explicitly selected, which shape the information processing in the human brain and distinguish it from other technical systems. The aspects and principles selected here are adaptation during the encoding of stimuli, the comparatively low signal transmission speed, the continuous formation and elimination of connections, and highly complex, partly chaotic, dynamics. The selection of these phenomena and properties has led to the development of a sensory unit that is capable of encoding mechanical stress into a series of voltage pulses by the use of a MOSFET augmented by AlScN. The circuit is based on a leaky integrate and fire neuron model and features an adaptation of the pulse frequency. Furthermore, the slow signal transmission speed of biological systems was the motivation for the investigation of a temporal delay in the feedback of the output pulses of a relaxation oscillator. In this system stable pulse patterns which form due to so-called jittering bifurcations could be observed. In particular, switching between different stable pulse patterns was possible to induce. In the further course of the work, the first steps towards time-varying coupling of dynamic systems are investigated. It was shown that in a system consisting of dimethyl sulfoxid and zinc acetate, oscillators can be used to force the formation of filaments. The resulting filaments then lead to a change in the dynamics of the oscillators. Finally, it is shown that in a system with chaotic dynamics, the extension of it with a memristive device can lead to a transient stabilisation of the dynamics, a behaviour that can be identified as a repeated pass of Hopf bifurcations

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Computation of electromagnetic fields in assemblages of biological cells using a modified finite difference time domain scheme. Computational electromagnetic methods using quasi-static approximate version of FDTD, modified Berenger absorbing boundary and Floquet periodic boundary conditions to investigate the phenomena in the interaction between EM fields and biological systems.

yesThere is an increasing need for accurate models describing the electrical behaviour of individual biological cells exposed to electromagnetic fields. In this area of solving linear problem, the most frequently used technique for computing the EM field is the Finite-Difference Time-Domain (FDTD) method. When modelling objects that are small compared with the wavelength, for example biological cells at radio frequencies, the standard Finite-Difference Time-Domain (FDTD) method requires extremely small time-step sizes, which may lead to excessive computation times. The problem can be overcome by implementing a quasi-static approximate version of FDTD, based on transferring the working frequency to a higher frequency and scaling back to the frequency of interest after the field has been computed. An approach to modeling and analysis of biological cells, incorporating the Hodgkin and Huxley membrane model, is presented here. Since the external medium of the biological cell is lossy material, a modified Berenger absorbing boundary condition is used to truncate the computation grid. Linear assemblages of cells are investigated and then Floquet periodic boundary conditions are imposed to imitate the effect of periodic replication of the assemblages. Thus, the analysis of a large structure of cells is made more computationally efficient than the modeling of the entire structure. The total fields of the simulated structures are shown to give reasonable and stable results at 900MHz, 1800MHz and 2450MHz. This method will facilitate deeper investigation of the phenomena in the interaction between EM fields and biological systems. Moreover, the nonlinear response of biological cell exposed to a 0.9GHz signal was discussed on observing the second harmonic at 1.8GHz. In this, an electrical circuit model has been proposed to calibrate the performance of nonlinear RF energy conversion inside a high quality factor resonant cavity with known nonlinear device. Meanwhile, the first and second harmonic responses of the cavity due to the loading of the cavity with the lossy material will also be demonstrated. The results from proposed mathematical model, give good indication of the input power required to detect the weakly effects of the second harmonic signal prior to perform the measurement. Hence, this proposed mathematical model will assist to determine how sensitivity of the second harmonic signal can be detected by placing the required specific input power

21st Century Nanostructured Materials

Nanostructured materials (NMs) are attracting interest as low-dimensional materials in the high-tech era of the 21st century. Recently, nanomaterials have experienced breakthroughs in synthesis and industrial and biomedical applications. This book presents recent achievements related to NMs such as graphene, carbon nanotubes, plasmonic materials, metal nanowires, metal oxides, nanoparticles, metamaterials, nanofibers, and nanocomposites, along with their physical and chemical aspects. Additionally, the book discusses the potential uses of these nanomaterials in photodetectors, transistors, quantum technology, chemical sensors, energy storage, silk fibroin, composites, drug delivery, tissue engineering, and sustainable agriculture and environmental applications