This project looks into the time evolution of a wave function within a two-dimensional quantum well. We start by solving the time-dependent Schrödinger equation for stationary states in a quantum well. Next, we express any wave function as a linear combination of stationary states, allowing us to understand their time evolution. Two methods are presented: one relies on decomposing the wave function into a basis of stationary states and the other on discretisation of the time-dependent Schrödinger equation, incorporating three-point formulas for derivatives. These approaches necessitate confronting intricate boundary conditions and require maintaining energy conservation for numerical accuracy. We further demonstrate the methods using a wave packet, revealing fundamental phenomena in quantum physics. Our results demonstrate the utility of these methods in understanding quantum systems, despite the challenges in determining stationary states for a given potential. This study enhances our comprehension of the dynamics of quantum states in constrained systems, essential for fields like quantum computing and nanotechnology.

This chapter focuses on the project of finding the potential for a given distribution of charges in a two-dimensional system, which does not possess any symmetrical properties, an extension of the cylindrical potential problem discussed in the previous chapter. Using a method of minimising a functional, specifically the Gauss–Seidel method of iterative minimisation, the Poisson’s equation is adjusted to a 2D case, neglecting one partial derivative in Cartesian coordinates. We subsequently derive a discretised form of the functional, leading to a multi-variable function, following which the problem can be solved using the Gauss–Seidel iterative method. The numerical method discussed here is the finite elements method (FEM), with an emphasis on the need for a specific sequence for updating values to optimise computation efficiency. The discussion sheds light on the importance of the uniqueness of solutions in electrostatic systems, thereby exploring a fundamental question in electrostatics. The concluding part of the chapter provides an outline of a numerical algorithm for the problem, suggesting potential modifications and points for further exploration.

This chapter revisits each of the design principles, summarizing and drawing connections between them. Many of the principles are based on empirical evidence from traditional learning environments; a discussion on the boundary conditions of the design principles explores the extrapolations of this evidence to training recognition skills in dynamic, high stakes environments. The chapter closes with a discussion of the contributions and challenges of augmented reality to training.

Most sand seas and dune fields exhibit clear spatial patterns of dune morphological type as well as variations in sediment thickness, dune size, crest length, and spacing. These patterns are the geomorphic expression of the factors that have controlled sand sea development in time and space. They reflect the self-organizing nature of dune systems as well as the external geomorphic and climatic environment (boundary conditions) in which the sand sea has evolved.

This chapter reviews the computer simulation of simple lattice models for uniaxial and biaxial nematic systems. Beyond being interesting in their own right for understanding the orientational properties of LCs, these models, e.g. the Lebwohl–Lasher one, are computationally unexpensive in relative terms, and provide a useful test bed for developing techniques for studying LCs that can then be employed also for off-lattice and atomistic models. Here the investigation of the orientational phase transition, assessing its type, as well as the identification of topological defects and the calculation of DNMR spectra in bulk and confined nematics (droplets, films) are discussed.

A simple introduction to the Metropolis Monte Carlo method with a discussion of the main types of boundary conditions. The location of phase transitions and the investigation of pair correlation functions, important for establishing the existence of long-range order in liquid crystal models is also introduced.

To model glacial triggering of earthquakes, it is necessary to obtain the spatio-temporal variation of glacial isostatic adjustment-induced stress during a glacial cycled. This can be computed efficiently using commercial Finite Element codes with appropriate modifications to include the important effects of ‘pre-stress advection’, ‘internal buoyancy’ and ‘self-gravity’. The modifications described in Wu (2004) are reviewed for incompressible and so-called materially compressible flat-earths. When the glacial isostatic adjustment-induced stress is superimposed on the background tectonic stress and overburden pressure, the time variation of earthquake potential at various locations in the Earth can be evaluated for any fault orientation. To model more complex slip and fault behavior over time, the three-stage Finite Element model approach of Steffen et al. (2014) is reviewed. Finally, selected numerical examples and their results from both modelling approaches are shown.

The knowledge gained in the previous two chapters leads to procedures for computing solutions to the Navier–Stokes equations in 2D and 3D. Chapter 6 explains the major components and functions of a typical Reynolds-averaged Navier–Stokes (RANS) code, including the modeling of turbulence in steady or unsteady flows. Convergence acceleration devices, including multigrid techniques, are explained. Finite-volume formulation and standard physical modeling for turbulence yields the RANS equations used in most computational fluid dynamics (CFD) codes directed toward compressible-flow aeronautical applications. By taking the reader through a RANS application step by step, this chapter illustrates the process that an informed CFD user needs to know for applying a typical code of this genus to aerodynamic design. Two practical cases of transonic flow over an airfoil – one in steady flow and the other in unsteady buffeting flow – demonstrate execution of the workflow. Computing a Mach sweep across the entire transonic regime, the steady-flow example exhibits the nonlinear phenomenon of shock stall. Mastering this chapter makes the student a reasonably well-informed CFD user who understands how to carry out a sensitivity analysis to demonstrate CFD due diligence.

The aerodynamics of airplanes designed before and during World War II springs from linear potential theory (discussed in Chapter 3) together with empirical data and lessons learned from previous airplane designs. Jet engines and rocket propulsion enabled vehicles to fly much faster. This uncovered high-speed aerodynamic phenomena that must be understood for the successful design of airplanes capable of trans- and supersonic flight and of space vehicles. This chapter presents the numerical methods employed in computational fluid dynamics (CFD) to treat shock waves. A complete recipe for inviscid nozzle flow is given, with accompanying tutorial software. Mature tools are now standard in the form of industrial-strength CFD codes. A perusal of the user manual for any one of them shows many options and functions. A basic understanding of the theory is needed for the user to set up the code properly for the intended case. While not focusing on constructing such a CFD code, this chapter lays the foundations for training the reader to become an "informed user" of these codes by learning CFD "due diligence." It spells out CFD fundamentals such as constructing the numerical flux, artificial dissipation, approximate Riemann, high-resolution schemes. explicit and implicit time integration, and convergence to steady state.

Previous chapters examined the topics of equilibrium, compatibility, strain–displacement relations, and stress–strain relations. When these elements are combined, we can form up different complete sets of governing differential and algebraic equations. In order to solve those sets of equations we must also specify the conditions that arise from having known loads or geometric constraints. These are called the boundary conditions for the problem.In this chapter we will examine some of the choices we have for formulating complete sets of the governing equations and how those governing equations can be combined with appropriate boundary conditions to solve for the stresses and deformations. We will also discuss the principle of Saint-Venant, which gives us some flexibility in how we specify the boundary conditions. Finally, we will also show how structural analysis problems can be expressed in terms of algebraic matrix–vector equations, which are the counterparts of the governing differential/algebraic equations. A classical deformable body problem, Navier's table problem, will be used as an example of these purely algebraic methods.

Partial differential equations whose solution specifies the elastic response of a loaded body are summarized. If all boundary conditions are given in terms of tractions, the boundary-value problem can be specified entirely in terms of stresses. The governing differential equations are then the Cauchy equations of equilibrium and the Beltrami–Michell compatibility equations. If some of the boundary conditions are given in terms of the displacements, the boundary-value problem is formulated in terms of the displacement components through the Navier equations of equilibrium. The boundary conditions can be expressed in terms of displacements, or in terms of displacement gradients. Due to the linearity of all equations and boundary conditions, the principle of superposition applies in linear elasticity. The semi-inverse method of solution and the Saint-Venant principle are introduced and discussed. The solution procedure is illustrated in the analysis of the stretching of a prismatic bar by its own weight, thermal expansion of a compressed prismatic bar, pure bending of a prismatic bar, and torsion of a prismatic rod with a circular cross section.

In this chapter Maxwell’s equations are described and common ways to solve them analytically are discussed. The equations imply certain properties of matter with which it interacts and full solutions that describes this behavior analytically are provided from first principles. The chapter shows specifically that one can derive very fundamental properties with simple calculations. Furthermore, the concept of inductance and capacitance are highlighted by reference to their duality. Various high-speed phenomena are studied in some detail with particular attention to the current distributions induced by the magnetic field.

This chapter explains how the mathematical models from Chapter 4 are implemented and integrated to form a full simulator. To this end, we introduce data structures to represent fluid behavior, the reservoir state, boundary conditions, source terms, and wells. We then explain in detail how the two-point flux approximation (TPFA) scheme is implemented in MRST for general unstructured grids. We also outline the basic solver used to compute time-of-flight and tracer partitions. We end the chapter by presenting a few examples that demonstrate how to set up simulations in MRST and set appropriate boundary conditions, source terms, or well models. The examples include the famous quarter-five spot problem, a corner-point grid with four intersecting faults, and a model of a shallow-marine reservoir (SAIGUP).

The chapter introduces you to mathematical modeling of flow in porous media. We start by explaining Darcy's law, which together with conservation of mass comprises the basic models for single-phase flow. We then discuss various special cases, including incompressible flow, constant compressibility, weakly compressible flow, and ideal gases. We then continue to discuss additional equations required to close the model, including equations of state, boundary and initial conditions. Flow in and out of wells take place on a smaller spatial scale and is typically modeled using special analytical submodels. We outline basic inflow–performance relationships for the special cases of steady and pseudo-steady radial flow, and develop the widely used Peaceman well model. We also introduce streamlines, time-of-flight, and tracer partitions that all can be used to understand flow patterns better. Finally, we introduce basic finite-volume discretizations, including the two-point flux approximation method, and show how such schemes can be implemented very compactly in MATLAB if we introduce abstract, discrete differentiation operators that are agnostic to grid geometry and topology.