A note on mixed boundary value problems involving triple trigonometrical series

This study was motivated by the two-dimensional hydrodynamic slamming problem of a steep wave hitting a vertical wall. The fundamental problem considers dual impact on the wall at the lower and upper regions resembling the impact of a wave at the time of its breaking. The solution method results into a mixed-boundary value problem that involves a triplet of trigonometrical series which, to the author’s best knowledge, has not been investigated in the past. The formulation of the mixed-boundary value problem is generic and could be used in different fields as well

Three-dimensional steep wave impact on a vertical plate with an open rectangular section

The present study treats the three-dimensional hydrodynamic slamming problem on a vertical plate subjected to the impact of a steep wave moving towards the plate with a constant velocity. The problem is complicated significantly by assuming that there is a rectangular opening on the plate which allows a discharge of the liquid. The analysis is conducted analytically assuming linear potential theory. The examined configuration determines two boundary value problems with mixed conditions which fully are taken into account. The mathematical process assimilates the plate with a degenerate elliptical cylinder allowing the employment of elliptical harmonics that ensure the satisfaction of the free-surface boundary condition of the front face of the steep wave, away from the plate. This assumption leads to an additional boundary value problem with mixed conditions in the vertical direction. The associated problem involves triple trigonometrical series and it is solved through a transformation into integral equations. To tackle the boundary value problem in the vertical direction a perturbation technique is employed. Extensive numerical calculations are presented as regards the variation of the velocity potential on the plate at the instant of the impact which reveals the influence of the opening. The theory is extended to the computation of the total impulse exerted on the plate using pressure-impulse theory

Three-dimensional steep wave impact on a vertical cylinder

In the present study we investigate the 3-D hydrodynamic slamming problem on a vertical cylinder due to the impact of a steep wave that is moving with a steady velocity. The linear theory of the velocity potential is employed by assuming inviscid, incompressible fluid and irrotational flow. As the problem is set in 3-D space, the employment of the Wagner condition is essential. The set of equations we pose, is presented as a mixed boundary value problem for Laplace's equation in 3-D. Apart from the mixed-type of boundary conditions, the problem is complicated by considering that the region of wetted surface of the cylinder is a set whose boundary depends on the vertical coordinate on the cylinder up to the free-surface. We make some simple assumptions at the start but otherwise we proceed analytically. We find closed-form relations for the hydrodynamic variables, namely the time dependent potential, the pressure impulse, the shape of the wave front (from the contact point to beyond the cylinder) and the slamming force

Algorithms for classic dual trigonometric equations

AbstractThe singular integral solutions to certain classic dual trigonometric equations provided by the formulas of Tranter and Bablojan are reduced to algorithms. A preliminary Fourier analysis is made of the data, and computational rules are derived by the systematic reduction of the singular integrals for each ordinary Fourier component of the data. Extensive numerical testing provides evidence for the correctness of both the original solutions and the resulting algorithms. The listing of programs in ANSI FORTRAN to implement the algorithms is appended

On quadruple integral equations involving trigonometric kernels

A general technique is developed for the solution of quadruple integral equations involving trigonometric kernels. Four such sets are solved explicitly. Application is made to the problem of three-collinear cracks in linear plane elasticity

Real Analysis, Harmonic Analysis and Applications

[no abstract available

Theory and application of the adjoint method in geodynamics and an extended review of analytical solution methods to the Stokes equation

The initial condition problem with respect to the temperature distribution in the Earth's mantle is Pandora's box of geodynamics. The heat transport inside the Earth follows the principles of advection and conduction. But since conduction is an irreversible process, this mechanism leads to a huge amount of information getting lost over time. Due to this reason, a recovery of a detailed state of the Earth's mantle some million years ago is an intrinsically unsolvable problem. In this work we present a novel mathematical method, the adjoint method in geodynamics, that is not capable of solving but of circumventing the presented initial condition problem by reformulating this task in terms of an optimisation problem. We are aiming at a past state of the Earth's mantle that approaches the current and thus, observable state over time in an optimal way. To this end, huge computational resources are needed since the 'optimal' solution can only be found in an iterative process. In this work, we developed a new general operator formulation in order to determine the adjoint version of the governing equations of mantle flow and applied this method to the high-resolution numerical mantle circulation code TERRA. For our models, we used a global grid spacing of approx. 30 km and more than 80 million mesh elements. We found a reconstruction of the Earth's mantle at 40 Ma that is, with respect to our modelling parameters, consistent with today's observations, gathered from seismic tomography. With this published fundamental work, we are opening the door to a variety of future applications, e.g. a possible incorporation of geological and geodetic data sets as further constraints for the model trajectory over geological time scales. Where high-resolution numerical models and even the implementation of inversion schemes have become feasible over the past decades due to increasing computational resources, in the community there is still a high demand for analytical solution methods. Restricting the physical parameter space in the governing equations, e.g. by only allowing for a radial varying viscosity, it can be shown that in some cases, the resulting simplified equations can even be solved in a (semi-)analytical way. In other words, in these simplified scenarios, no large scale computational resources or even high-performance clusters are needed but the solution for a global flow system can be determined in minutes even on a standard computer. Besides this apparent advantage, analytical and numerical solutions can even go hand-in-hand since numerical computer codes may be tested and benchmarked by means of these manufactured solutions. Here, we spend a large portion of this work with a detailed derivation of these analytical approaches. We basically start from scratch, having the intention to cover all possible traps and pitfalls on the way from the governing equations to their solutions and to provide a service to future scientists that are stuck somewhere in the middle of this road. Besides the derivation, we also present in detail how such an analytical approach can be used as a benchmark for a high-resolution mantle circulation code. We applied this theory to the prototype for a new high-performance mantle convection framework being developed in the Terra-Neo project and published the results along with a small portion of the derived theory. In an additional chapter of this work, we focus on a detailed analysis of the current state of the Earth's gravitational field that is measured in an unimaginably accurate way by the recent satellite missions CHAMP, GRACE and GOCE. The origin of the link of our work to the gravitational field also lies in the analytical solution methods. It can be shown that due to the effect of flow induced dynamic topography, the Earth's gravity field is highly sensitive to the viscosity profile in the Earth's mantle. We show that even without using any other external knowledge or data set, the gravitational field itself restricts the possible choices for the Earth's mantle viscosity to a well-defined parameter space. Furthermore, in the course of these examinations, we found that mantle processes are not capable of explaining the short wavelength signals in the observed gravity field at all, even with the best-fitting viscosity profile. To this end, we developed a simple crustal model that is only based on topographic data (ETOPO) and the principle of isostasy and showed that even with this very basic approach we can explain the majority of short length-scale features in the observed gravity signal. Finally, in combination with a (simple, static and analytic) mantle flow model based on a density field derived from seismic topography and mineralogy, we found a nearly perfect fit of modelled and observed gravitational data throughout all wavelengths under consideration (spherical harmonic degree and order up to l=100)