Gradient Preserved And Artificial Neural Network Method For Solving Heat Conduction Equations In Double Layered Structures

Layered structures have appeared in many engineering systems such as biological tissues, micro-electronic devices, thin  lms, thermal coating, metal oxide semiconductors, and DNA origami. In particular, the multi-layered metal thin  lms, gold-coated metal mirrors for example, are often used in high-powered infrared-laser systems to avoid thermal damage at the front surface of a single layer  lm caused by the high-power laser energy. With the development of new materials, functionally graded materials are becoming of more paramount importance than materials having uniform structures. For instance, in semiconductor engineering, structures can be synthesized from di erent polymers, which result in various values of conductivity. Analyzing heat transfer in layered structure is crucial for the optimization of thermal processing of such multi-layered materials. There are many numerical methods dealing with heat conduction in layered structures such as the Immersed Interface Method, the Matched Interface Method, and the Boundary Method. However, development of higher-order accurate stable nite di erence schemes using three grid points across the interface between layers for variable coe cient case is mathematically challenging. Having three grid points ensures that the nite di erence scheme leads to a tridiagonal matrix that can be solved easily using the Thomas Algorithm. But extension of such methods to higher dimensions is very tedious. Recently there have been some solution to such complex systems with the use of neural networks, that can be easily extended to higher dimensions. For the above purposes, in this dissertation, we  rst develop a gradient preserved method for solving heat conduction equations with variable coe cients in double layers. To this end, higher-order compact  nite di erence schemes based on three grid points are developed. The  rst-order spatial derivative is preserved across the interface. Unconditional stability and convergence with O(  2 + h4) are analyzed using the discrete energy method, where   and h are the time step and grid size, respectively. Numerical error and convergence rates are tested in an example. We then present an arti cial neural network (ANN) method for solving the parabolic two-step heat conduction equations in double-layered thin  lms exposed to ultrashort-pulsed lasers. Convergence of the ANN solution to the analytical solution is theoretically analyzed using the energy method. Finally, both developed methods are applied for predicting electron and lattice temperature of a solid thin  lm padding on a chromium  lm exposed to the ultrashort-pulsed lasers. Compared with the existing results, both methods provide accurate solutions that are promising

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

RIACS

Topics considered include: high-performance computing; cognitive and perceptual prostheses (computational aids designed to leverage human abilities); autonomous systems. Also included: development of a 3D unstructured grid code based on a finite volume formulation and applied to the Navier-stokes equations; Cartesian grid methods for complex geometry; multigrid methods for solving elliptic problems on unstructured grids; algebraic non-overlapping domain decomposition methods for compressible fluid flow problems on unstructured meshes; numerical methods for the compressible navier-stokes equations with application to aerodynamic flows; research in aerodynamic shape optimization; S-HARP: a parallel dynamic spectral partitioner; numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains; application of high-order shock capturing schemes to direct simulation of turbulence; multicast technology; network testbeds; supercomputer consolidation project

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp

Refresher course in maths and a project on numerical modeling done in twos

These lecture notes accompany a refresher course in applied mathematics with a focus on numerical concepts (Part I), numerical linear algebra (Part II), numerical analysis, Fourier series and Fourier transforms (Part III), and differential equations (Part IV). Several numerical projects for group work are provided in Part V. In these projects, the tasks are threefold: mathematical modeling, algorithmic design, and implementation. Therein, it is important to draw interpretations of the obtained results and provide measures (Parts I-IV) how to build confidence into numerical findings such intuition, error analysis, convergence analysis, and comparison to manufactured solutions. Both authors have been jointly teaching over several years this class and bring in a unique mixture of their respective teaching and research fields

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

Computational Multiscale Methods

Many physical processes in material sciences or geophysics are characterized by inherently complex interactions across a large range of non-separable scales in space and time. The resolution of all features on all scales in a computer simulation easily exceeds today's computing resources by multiple orders of magnitude. The observation and prediction of physical phenomena from multiscale models, hence, requires insightful numerical multiscale techniques to adaptively select relevant scales and effectively represent unresolved scales. This workshop enhanced the development of such methods and the mathematics behind them so that the reliable and efficient numerical simulation of some challenging multiscale problems eventually becomes feasible in high performance computing environments