Wavelet Numerical Method With Some Applications To Marine Hydrodynamics

Wavelets analysis was proposed in 1980s, it has been rapidly used in various subjects, for its time-frequency localized function. Wavelets analysis came from signal analysis area, now it's playing an important role in signal analysis and process area. Because wavelet basis function have favorable mathematical properties, so wavelet method can make some contributions in numerical solution of partial differential equations. Because Wavelets provide basis with finite support, so it capture local structure, even the singular solutions. The main idea of wavelet analysis lies in multi-resolution, which can make functions with mixed scales'information project to separated scales, further more; each wavelet coefficient is corresponding to each grid in spatial position. Fourier analysis lacks the latter property. These properties make wavelet method have advantages in simulating complicated flow.The wavelet method in partial differential equation can be divided into two types: wavelet projection methods and wavelet collocation methods. We study wavelet collocation method in this paper, and we use it to solve nonlinear time evolution equation. As we known, nonlinear time evolution equations often have singular solution, which make difference schemes complicated and empirical. So people find it's not convenient to use them. Wavelet adaptive method can set a function's approximation accuracy, which corresponding to a threshold for wavelet coefficients. The threshold can control the grids'number for approximation function on one particular grid. Two benefits are brought by this method, one is saving much computational amount, and the other is controlling error in global domain. The two benefits make numerical oscillation avoided. We calculate two differential equations with analytic solution to verify the the method's ability of controlling error, and then Burgers equations and its related equations is solved using wavelet collocation adaptive method, results show the method is effective.There are many problems with multi-scale property in marine hydrodynamics such as turbulence, shock and wave movement motivated by ship. Fourther more, intermittency is found in turbulence and discontinuity is found in shock, we will apply adaptive wavelet methods in these problems'calculations. We use wavelet collocation adaptive method to simulate turbulence, and we propose idea that turbulence's simulation should be focused on the flow itself. Turbulence coherent structure's simulated based on the threshold for wavelet coefficients, which is the statics value of flow data, no artificial parameter. We take two dimensional turbulence for example, three vortex merging process is calculated, physical pictures are obtained. Results show that wavelet methods do capture the flow's main characters, the vorticity reconstructed by scale coefficients and significant wavelet coefficients can capture enstrophy more than 99%. The second viscosity mode is necessary for shock numerical simulation, and we use this model to calculate shock by LDQ method to calculate the derivatives, which is equivalent to interpolating wavelet. We obtain non-numerical oscillation and high resolution results. The second viscosity model and interpolating wavelet may help us to understand shock and calculate them easier.Since ship form optimization based on wave resistance has significant meaning in engineering, we try to study this topic based on wavelet method. We use Michell resistance theory and Wigley ship model to explain the wavelet method for ship form optimization. Wavelet series are used to express ship surface under water, and compressed expression is obtained using wavelet's high compression function. We get the resistance formula constituted by wavelet coefficients with the the wavelet's property of orthogonality. We take wavelet coefficients as optimization variables, then we get the formula for optimalization, volume is mantained when the coefficients are changed, thus the optimalization with constraint is coverted to optimalization without constraint, which simplifies the optimalization. We study the resistance distribution on each scale and location, and choose the wavelet coefficients on large scale as optimization variables, to reduce the number of optimal variables. We use the genetic algorithms, and get the optimal ship form.