Over the past few decades, dynamic solid mechanics has become a major field of interest in industrial applications involving crash simulation, impact problems, forging and many others to be named. These problems are typically nonlinear due to large deformations (or geometrical nonlinearity) and nonlinear constitutive relations (or material nonlinearity). For this reason, computer simulations for such problems are of practical importance. In these simulations, the Lagrangian formulation is typically used as it automatically satisfies the mass conservation law. Explicit numerical methods are considered to be efficient in these cases. Most of the numerical methods employed for such simulations are developed from the equation of motion (or momentum balance principle). The use of low-order elements in these numerical methods often exhibits the detrimental locking phenomena in the analysis of nearly incompressible applications, which produces an undesirable effect leading to inaccurate results. Situations of this type are usual in the solid dynamics analysis for rubber materials and metal forming processes. In metal plasticity, the plastic deformation is isochoric (or volume-preserving) whereas, the compressible part is due only to elastic deformation. Recently, a new mixed formulation has been established for explicit Lagrangian transient solid dynamics. This formulation, involving linear momentum, deformation gradient and total energy, results in first order hyperbolic system of equations. Such conservation-law formulation enables stresses to converge at the same rate as velocities and displacements. In addition, it ensures that low order elements can be used without volumetric locking and/or bending difficulty for nearly incompressible applications. The new mixed formulation itself shows a clear advantage over the classical displacement-based formulation, due to its simplicity in incorporating state-of-the-art shock capturing techniques. In this research, a curl-preserving cell centred finite volume computational methodology is presented for solving the first order hyperbolic system of conservation laws on quadrilateral cartesian grids. First, by assuming that the approximation to the unknown variables is constant within each cell. This will lead to discontinuities at cell edges which will motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of the numerical flux function. Unfortunately, the accuracy is severely undermined by an excess of numerical dissipation. In order to alleviate this, it is vital to introduce a linear reconstruction procedure for enhancing the accuracy of the scheme. However the second-order spatial method does not prohibit spurious oscillation in the vicinity of sharp gradients. To circumvent this, a nonlinear slope limiter will then be introduced. It is now possible to evolve the semi-discrete evolutionary system of ordinary equations in time with the aid of the family of explicit Total Variation Diminishing Runge Kutta (TVD-RK) time marching schemes. Moreover, a correction procedure involving minimisation algorithm for conservation of the total angular momentum is presented. To this end, a number of interesting examples will be examined in order to demonstrate the robustness and general capabilities of the proposed approach.

Controlled cooling is an important part of steel production industry that affects the properties of the outcome steel. Many of the researches done in controlled cooling are experimental. Due to progress in the numerical techniques and high cost of experimental works in this field the numerical work seems more feasible. Heat transfer analysis is the necessary element of successful controlled cooling and ultimately achievement of novel properties in steel. Heat transfer on the surface of the plate normally contains different regimes such as film boiling, nucleate boiling, transition boiling and radiation heat transfer. This makes the analysis more complicated. In order to perform the heat transfer analysis often empirical correlations are being used. In these correlations the velocity and pressure within the fluid domain is involved. Therefore in order to obtain a better understanding of heat transfer process, study of hydrodynamics of the fluid becomes necessary. Circular jet due to its high efficiency has been used vastly in the industry. Although some experimental studies of round jet arrays have been done, yet the characteristics of a single jet with industrial geometric and flow parameters on the surface of a flat plate is not fully understood. Study of hydrodynamics of the jet impingement is the first step to achieve better understanding of heat transfer process. Finite element method as a popular numerical method has been used vastly to simulate different domains. Traditional approaches of finite element method, Lagrangian and Eulerian, each has its own benefits and drawbacks. Lagrangian approach has been used widely in solid domains and Eulerian approach has been widely used in fluid fields. Jet impingement problem, due to its unknown free surface and the change in the boundary, falls in the category of special problems and none of the traditional approaches is suitable for this application. The Arbitrary Lagrangian Eulerian (ALE) formulation has emerged as a technique that can alleviate many of the shortcomings of the traditional Lagrangian and Eulerian formulations in handling these types of problems. Using the ALE formulation the computational grid need not adhere to the material (Lagrangian) nor be fixed in space (Eulerian) but can be moved arbitrarily. Two distinct techniques are being used to implement the ALE formulation, namely the operator split approach and the fully coupled approach. This thesis presents a fully coupled ALE formulation for the simulation of flow field. ALE form of Navier-Stokes equations are derived from the basic principles of continuum mechanics and conservation laws in the fluid. These formulations are then converted in to ALE finite element equations for the fluid flow. The axi-symmetric form of these equations are then derived in order to be used for jet impingement application. In the ALE Formulation as the mesh or the computational grid can move independent of the material and space, an additional set of unknowns representing mesh movement appears in the equations. Prescribing a mesh motion scheme in order to define these unknowns is problem-dependent and has not been yet generalized for all applications. After investigating different methods, the Winslow method is chosen for jet impingement application. This method is based on adding a specific set of partial differential Equations(Laplace equations) to the existing equations in order to obtain enough equations for the unknowns. Then these set of PDEs are converted to finite element equations and derived in axi-symmetric form to be used in jet impingement application. These equations together with the field equations are then applied to jet impingement problem. Due to the number of equations and nonlinearity of the field equations the solution of the problem faces some challenges in terms of convergence characteristics and modeling strategies. Some suggestions are made to deal with these challenges and convergence problems. Finally the numerical treatment and results of analyzing hydrodynamics of the Jet Impingement is presented. The work in this thesis is confined to the numerical simulation of the jet impingement and the specifications of an industrial test setup only have been used in order to obtain the parameters of the numerical model. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

Classical mechanics is the branch of physics concerned with describing the motion of bodies. The subject is based on three simple axioms relating forces and movement. These axioms were first postulated by Newton in the 17th century and are known as his three laws of motion. Lagrangian mechanics is a restatement of the Newtonian formulation. It deals with energy quantities and paths-of-motion instead of forces. This often makes it simpler to use when working with non-trivial mechanical systems. In this thesis, we use the Lagrangian method to model two such systems; A rotating torus and a variant of the classical double pendulum. It soon becomes clear that the complexity of these systems make them difficult to attack by hand. For this reason, we take a computer-based approach. We use a software-package called Sophia which is a plug-in to the computer algebra system Maple. Sofia was developed at the Department of Mechanics at KTH for the specific purpose of modeling mechanical problems using Lagrange’s method. We demonstrate that this method can be successfully applied to the analysis of motion of complex mechanical systems. The complete equations of motion are derived in a symbolic form and then integrated numerically. The motion of the system is finally visualized by means of 3D graphics software Blender.

No description available.

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.

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We present a framework for modeling multibody systems based on the method of natural coordinates and Lagrange's equation of the first kind, resulting in a system of Differential-Algebraic Equations (DAEs). The C++ package DAETS (DAEs by Taylor Series), a robust high-index DAE solver, is utilized to solve the models. The simulation process is straightforward, with no need to derive equations of motion directly. Instead, the user supplies a Lagrangian, kinematic constraints, and if applicable, a dissipation function and external forces. A corresponding system of DAEs is formed by computing the required derivatives via automatic differentiation. DAETS primarily uses Cartesian coordinates as variables, eliminating angles and the associated trigonometric functions, which results in simplified models. Furthermore, DAETS provides direct access to the position/velocity data of any desired points or vectors as output, facilitating post-processing tasks, such as visualization. The main focus of this thesis is on establishing the viability of our framework through case studies. We simulate seven multibody systems and compare our results with those of reference models developed in the Simulink environment of MATLAB. A detailed account of the modeling process is given for each system, demonstrating the ease and intuitiveness of our approach. We also provide, from both DAETS and Simulink, the time history plots of several position coordinates to allow for direct comparison. Finally, we compute two types of errors over time. Our findings show that the results of DAETS match those of the reference models under different error tolerances for the studied systems, indicating that our framework is capable of simulating a wide variety of mechanisms with a superb degree of accuracy. / Thesis / Master of Science (MSc)

While the operating conditions are the main factors that influence engine design, it is important to understand ignition in any potential design to ensure reliable light-ability. Ignition probability maps can be generated, either experimentally or numerically, to inform design of ignition mechanisms. Recent models have been proposed to estimate ignition probability using non-reacting computational fluid dynamic (CFD) simulations. These models have not been applied to scramjet flame holding cavities. A qualitative model is described that uses tracer particles that probe CFD data and are removed when the conditions are adverse to flame survivability. The parameters that influence ignition are investigated by changing the criteria to define the flammable region. A quantitative model is developed based on a frozen flow assumption and the assumption that a region exists such that the geometry can be considered ignited if a flame is able to be propagated to this region. A virtual flame begin in this "ignition region" and propagates backwards in time to all the cells that would be successful if forward time was used. This model is implemented with an Eulerian and a Lagrangian scheme (IMIT and LIMIT, respectively). The results are compared to a previous coldlow model, I-CRIT-LES, on a low speed, lifted jet geometry and a supersonic cavity geometry. The models generate similar results on the jet case. A diffusion-like effect in IMIT allows the virtual flame to propagate over streamlines and into cells that the flame should not be able to reach. Thus the cavity ignition map generated by IMIT overpredicts ignition. The diffusion-like problem is solved by using particles following the streamlines. Therefore, LIMIT results match the qualitative experimental data in the cavity better than the other models.

No description available.

No description available.