181 research outputs found
Energy-conserving spectral element schemes based on Lagrangian dynamics: Investigating discrete energy conservation
No full textPhysical systems in the continuous domain are often solved using computer-aided software because of their complexity. Preserving the physical quantities from the continuous domain in the discrete domain is therefore of utmost importance. There is however a broad range of techniques that can accomplish the translation between the continuous and discrete domain, e.g. finite difference, volume and element techniques, fourth order Runge-Kutta or Störmer-Verlet to name a few. Accompanied with the aforementioned come strengths and weaknesses but have the common thread to try maintaining the physical behaviour of the continuous system closely. The mimetic spectral element technique is used to develop an energy-conserving spectral element scheme through a Lagrangian formulation. This new formulation of the mimetic spectral element technique allows for solving time-dependent problems and the simple harmonic oscillator serves as the sample problem in this thesis. The solution has been derived from Lagrangian mechanics in a variety of ways. Discrete Lagrangian formulations have been investigated at rst and their respective equations of motions have been tested against the exact solution of the simple harmonic oscillator. This method achieved marching in time and no damping of the solution, yet energy was only bounded and not exactly conserved. The mimetic spectral element formulation of the Lagrangian formulation showed diculties when using variational analysis, i.e. boundary treatment in the future. Arbitrary domain mapping was among the possible solutions, but this formulation was found to be unreliable and unsuccessful. It was found that a more robust formulation, i.e. the spectral marching method, was most suitable. Throughout this thesis the focus was put on the conservation of energy using a Lagrangian formulation. Except the spectral scheme using arbitrary domain mapping, all schemes kept the energy of the system bounded, but no energy was conserved up to machine precision. Using arbitrary domain mapping, the energy seemed to grow over time.Aerospace Engineerin
Springing of ships waves
No full textThis thesis is the result of an investigation of the assumptions underlying the general applied method for the calculation of springing of ships in waves, which has been proposed by the author some decade ago. It has been found that, contrary to the general practice in seakeeping research, the frequency response method can not be applied for the determination of the springing response to random wave loading. Other assumptions regarding the hydrodynamic springing excitation and 2-node vertical main hull response have been confirmed. - In moderate sea states springing is linear with respect to wave height. - The longitudinal distribution of the damping may be assumed to be proportional to the distribution of the mass, the stiffness or a combination of them. The normal mode method can therefore be applied. - All hydrodynamic forces due to the vibration of the ship should be taken into account, i.e. vibratory induced buoyancy, as well as damping and mass forces cannot be neglected. Uncertainties in the springing calculation method have been described: - the damping coefficient of the 2-node vertical vibration and its longitudinal distribution - the applicability of the present striptheory for the calculation of the distribution of wave exciting forces along the length of the ship in relatively short waves - the parameters associated with the correction to the frequency response method for the calculation of the springing response to random wave loading. The conclusions are based on results of analysis of full scale as well as model experiments, theoretical calculations and time domain computer simulations. Finally, a method is presented to account for the deficiencies in the frequency response method in the calculation of springing of ships in waves.Mechanical, Maritime and Materials Engineerin
Advancing the Mimetic Spectral Element Method: Towards Continuum Mechanics Applications
No full textMimetic discretisation techniques are a growing field in computational physics research. Among these techniques, the recently developed mimetic spectral element method allows for exact discretisation of metric independent relations. This has been proven numerically in various mixed formulations, for instance the mixed velocity-vorticity-pressure formulation for Stokes flow, where mass conservation was point-wise strongly satisfied by the solution in the computational domain. Another example is the mixed stress-displacement formulation for the linear elasticity equations, where the balance law of linear momentum was point-wise strongly satisfied as well. A recent extension to a hybrid method leads to additional attractive features, such as the ability to decompose a large part of the computation of the solution into smaller problems. The aim of the research is to find a formulation for linear elasticity that is hybridisable while strongly satisfying conservation of linear and angular momentum as well, where the combination of linear momentum conservation and symmetry of the stress tensor is equivalent to angular momentum conservation. The proposed formulation has a mixed basis of both primal and algebraic dual nodal and edge basis functions. It fulfils the requirements as it is shown to be hybridisable, to satisfy point-wise linear momentum, and the discrete representation of the stress tensor is point-wise symmetric, hence angular momentum conservation is point-wise satisfied as well. The thesis furthermore functions as an overview of the method applied to elliptic problems, showing the results for previous formulations, and as a starting point for the next steps towards applying the method to fluids. A first step is proposed on extending the new formulation to a Stokes flow formulation with the stress as primary unknown, aimed at satisfying both linear and angular momentum conservation as well as mass conservation.Aerospace Engineerin
Mimetic Discretization of the High Frequency Helmholtz Equation using Hermite Interpolating Polynomials
No full textNumerical modelling of high frequency waves is a complex and challenging area. Although the underlying equation seems simple, −∆φ − k2φ = f, the numerical challenges are not. This time harmonic wave equation is known as the Helmholtz equation. The main challenge to be studied is the pollution error, which is the difference between the actual and numerical wavenumber. Due to this error, the solution of most numerical methods deteriorates rapidly when increasing the wavenumber. So far, either problem specific solutions have been found or solutions which require knowledge of the solution itself. In this thesis a numerical method is developed to approach this challenge. A mimetic discretization of the Helmholtz equation using C1-continuous Hermite interpolating polynomials is developed to model the Helmholtz equation in 2 dimensions. Mimetic theory is briefly introduced after which the Hermite polynomials are defined and analysed for their interpolating properties. A least-squares variational problem is defined and discretized using 49 basis function at its lowest order. The model is verified using a single sinusoidal wave after whichplane wave problems are solved and analysed for their wavenumber-dependence. A diffraction and interference problem is set-up and compared to analytical solutions. The proposed method shows no significant advantages over the use of C0-continuous Lagrange polynomials in terms of effectiveness. Both methods show deterioration at the same wave numbers and equal polynomial order. The Hermite polynomials require less unknowns to reach the same accuracy. In short, a p-accurate solution is found using Hermite polynomialsat the cost of a p − 1 model using Lagrange polynomials. This benefit in efficiency does not outweigh the additional effort and boundary information necessary to construct the problem. However, the variety of coefficients offer an opportunity for further research to find relations to the numerical wavenumber in the search of a wavenumber conserving discretization.Aerospace Engineerin
The hydrodynamic forces and ship motions in waves
No full textMechanical, Maritime and Materials Engineerin
Nonlinear Behaviour of Fast Monohulls in Head Waves
No full textMechanical Maritime and Materials Engineerin
A Finite Element Method with Special Edge-Polynomials for Advection Problems
No full textThe field of Computational Fluid Dynamics (CFD) is constantly finding new ways to improve simulation results. One of the large challenges in CFD, is advection. Advection dominated flows are governed by hyperbolic equations, which pose severe stability issues. Stabilized methods usually add significant numerical diffusion, which alters the solution. Another issue are discontinuities in solutions, such as found in shock waves and acoustics. Besides solving these issues, conservative properties have to be satisfied at all costs. In order to further improve results, new methods are still developed today. The mathematical tools to do such developments are also further expanded. For instance, in the last decades the branch of differential geometry has steadily grown. This branch of mathematics can be extremely useful, since a clear distinction is made between geometric depend and geometric independent operators. These operators become more in use for numerical methods. Of all the available operators, the Lie derivative is important for advection. In junction with newly developed edge polynomials, a scheme for advection is created. The new edge polynomials are special, as these partly scale with a variable integrated over the cell and partly scale with the values on the cell boundaries. In order to do this, one polynomials integrates to 1 over the cell, while the other polynomials do not contribute, hence integrate to 0 over a cell. This gives several benefits. Conservation is clearly defined in the cells and not in points, hence conservation is locally defined. Also an extinction is made between conservative values and fluxes. One can also use the property of not contributing to conservation by correcting point values for stabilization or improvement of the solution. These polynomials are used within a Galerkin Finite Elements (FE) framework. Besides correction methods, upwind methods can be used as well. The approach is tested on several equation sets. On a simple linear advection problem, excellent results are found. Four test cases are tried, from a sine wave, two discontinuous functions and a hat function. With the proposed polynomials, the shapes are conserved extremely well, also in comparison to widely used schemes. On the inviscid Burgers’ equations, the stabilization methods become of importance. In the inviscid Burgers’ equation, initially smooth problems can become discontinuous. When a discontinuity forms, several issues can occur. For one, spurious oscillations can become apparent, or the velocity of the discontinuity can have errors. With several approaches, competitive results can be obtained. The discontinuity velocity is computed well, but spurious oscillation usually do occur. With several methods, the oscillations are suppressed. Not all approaches succeed in doing so. The third and final equation set to be tested, are the Euler equations. Although solutions can be found, all solutions contain errors. Several error sources are indicated, which should be addressed in further research. In comparison to a Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme, results need to be further improved to be on par. In conclusion, the suggested polynomials show potential. Several methods to improve solutions showed their usefulness. In linear advection, the new polynomials performed better compared to existing widely used methods. In non-linear advection, the results are on par with reference methods. For the Euler equations, further improvements are necessary. With further research, the benefits of such polynomials for numerical methods would become even more clear.Aerospace EngineeringAerodynamics, Wind Energy & Propulsio
Mimetic Discretizations with B-Splines: On the Construction of a Discrete Hodge Star Operator
No full textThis thesis introduces a higher-order numerical method for elliptic boundary value problems. The discretization method belongs to the class of mimetic discretizations, which translate as many of properties of the continuous problem to the discrete system, aiming to improve accuracy and reliability. The novelty lies in the application of B-splines as basis functions in a dual grid approach. B-splines or basis splines are piecewise polynomials with a certain degree of continuity between the polynomial pieces. Therefore, splines offer an attractive compromise between piecewise linear functions, commonly seen in finite element analysis, and the Lagrange polynomials from spectral element methods.Aerospace Engineering | Aerodynamics and Wind Energ
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