Benefit of poroelasticity for geothermal research

<p>For geothermal purposes (heat and electricity generation) it is necessary to have an aquifer from which the contained hot water can be lifted by drilling. The exchange of the hot water against some cooled off water has an effect on the surrounding material and displacement of the material has an influence on the pore pressure and the water. Poroelasticity can model these influencing effects by partial differential equations.</p><p>We want to apply poroelasticity in geothermal research by so-called multiscale modelling. Scaling functions and wavelets are constructed with the help of the fundamental solutions. A related method has been previously used for the Laplace, the Helmholtz and the d'Alembert equation (cf. [2],[4],[5]) as well as for the Cauchy-Navier equation, where the latter requires a tensor-valued ansatz (cf. [3]). We pursue this concept to develop such an approach for poroelasticity, where a fundamental solution tensor is known (cf. [1]).</p><p>The aim of this multiscale modelling is to convolve the constructed scaling functions with the data of the displacement $u$ and the pressure $p$. With this, we have the opportunity to visualize structures in the data that cannot be seen in the whole data. Especially, the difference of the convolution of two consecutive scaling functions is expected to reveal detail structures.</p><p>For the theoretical part, we can show that the scaling functions fulfill the property of an approximate identity. Furthermore, with numerical results we want to show the decomposition.</p><p><strong>References</strong></p><p>[1] M. Augustin: A method of fundamental solutions in poroelasticity to model the stress field in geothermal reservoirs, PhD Thesis, University of Kaiserslautern, 2015, Birkhäuser, New York, 2015.</p><p>[2] C. Blick, Multiscale potential methods in geothermal research: decorrelation reflected post-processing and locally based inversion, PhD Thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015.</p><p>[3] C. Blick, S. Eberle, Multiscale density decorrelation by Cauchy-Navier wavelets, Int. J. Geomath. 10, 2019, article 24.</p><p>[4] C. Blick, W. Freeden, H. Nutz: Feature extraction of geological signatures by multiscale gravimetry. Int. J. Geomath. 8: 57-83, 2017.</p><p>[5] W. Freeden, C. Blick: Signal decorrelation by means of multiscale methods, World of Mining, 65(5):304--317, 2013.<br><br></p>

Poroelastic aspects in geothermics

<p>The aspect of poroelasticity is anywhere interesting where a solid material and a fluid come into play and have an effect on each other. This is the case in many applications and we want to focus on geothermics. It is useful to consider this aspect since the replacement of the water in the reservoir below the Earth's surface has an effect on the sorrounding material and vice versa. The underlying physical processes can be described by partial differential equations, called the quasistatic equations of poroelasticity (QEP). From a mathematical point of view, we have a set of three (for two space and one time dimension) partial differential equations with the unknowns u (displacement) and p (pore pressure) depending on the space and the time.</p><p>Our aim is to do a decomposition of the data given for u and p in order that we can see underlying structures in the different decomposition scales that cannot be seen in the whole data.<br>For this process, we need the fundamental solution tensor of the QEP (cf. [1],[5]).<br>That means we assume that we have given data for u and p (they can be obtained for example by a method of fundamental solutions, cf. [1]) and want to investigate a post-processing method to these data. Here we follow the basic approaches for the Laplace-, Helmholtz- and d'Alembert equation (cf. [2],[4],[6]) and the  Cauchy-Navier equation as a tensor-valued ansatz (cf. [3]). That means we want to modify our elements of the fundamental solution tensor in such a way that we smooth the singularity concerning a parameter set τ=(τ<sub>x</sub>,τ<sub>t</sub>). <br>With the help of these modified functions, we construct scaling functions which have to fulfil the properties of an approximate identity.<br>They are convolved with the given data to extract more details of u and p.</p><p><strong>References</strong></p><p>[1] M. Augustin: A method of fundamental solutions in poroelasticity to model the stress field in geothermal reservoirs, PhD Thesis, University of Kaiserslautern, 2015, Birkhäuser, New York, 2015.<br>[2] C. Blick, Multiscale potential methods in geothermal research: decorrelation reflected post-processing and locally based inversion, PhD Thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015.<br>[3] C. Blick, S. Eberle, Multiscale density decorrelation by Cauchy-Navier wavelets, Int. J. Geomath. 10, 2019, article 24.<br>[4] C. Blick, W. Freeden, H. Nutz: Feature extraction of geological signatures by multiscale gravimetry. Int. J. Geomath. 8: 57-83, 2017.<br>[5] A.H.D. Cheng and E. Detournay: On singular integral equations and fundamental solutions of poroelasticity. Int. J. Solid. Struct. 35, 4521-4555, 1998.<br>[6] W. Freeden, C. Blick: Signal decorrelation by means of multiscale methods, World of Mining, 65(5):304-317, 2013.<br><br></p>

Holonomicity analysis of electromechanical systems

Electromechanical systems are described using state variables that contain electrical and mechanical components. The equations of motion, both electrical and mechanical, describe the relationships between these components. These equations are obtained using Lagrange functions. On the basis of the function and Lagrange - d’Alembert equation the methodology of obtaining equations for electromechanical systems was presented, together with a discussion of the nonholonomicity of these systems. The electromechanical system in the form of a single-phase reluctance motor was used to verify the presented method. Mechanical system was built as a system, which can oscillate as the element of physical pendulum. On the base of the pendulum oscillation, parameters of the electromechanical system were defined. The identification of the motor electric parameters as a function of the rotation angle was carried out. In this paper the characteristics and motion equations parameters of the motor are presented. The parameters of the motion equations obtained from the experiment and from the second order Lagrange equations are compared.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

ON THE REAL-VALUED GENERAL SOLUTIONS OF THE D’ALEMBERT EQUATION WITH INVOLUTION

We find all real-valued general solutions$f:S\rightarrow \mathbb{R}$of the d’Alembert functional equation with involution$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$for all$x,y\in S$, where$S$is a commutative semigroup and$\unicode[STIX]{x1D70E}~:~S\rightarrow S$is an involution. Also, we find the Lebesgue measurable solutions$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$of the above functional equation, where$\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$is a Lebesgue measurable involution. As a direct consequence, we obtain the Lebesgue measurable solutions$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$of the classical d’Alembert functional equation$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$for all$x,y\in \mathbb{R}^{n}$. We also exhibit the locally bounded solutions$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$of the above equations.

Simultaneous path planning and trajectory optimization for robotic manipulators using discrete mechanics and optimal control

SUMMARYIn robotics, path planning and trajectory optimization are usually performed separately to optimize the path from the given starting point to the ending point in the presence of obstacles. In this paper, path planning and trajectory optimization for robotic manipulators are solved simultaneously by a newly developed methodology called Discrete Mechanics and Optimal Control (DMOC). In DMOC, the Lagrange–d'Alembert equation is discretized directly unlike the conventional variational optimization method in which first the Euler–Lagrange equations are derived and then discretization takes place. In this newly developed method, the constraints for optimization of a desired objective function are the forced discrete Euler–Lagrange equations. In this paper, DMOC is used for simultaneous path planning and trajectory optimization for robotic manipulators in the presence of known static obstacles. Two numerical examples, applied on a DELTA parallel robot, are discussed to show the applicability of this new methodology. The optimal results obtained from DMOC are compared with the other state-of-the-art techniques. The difficulties and problems associated in using the DMOC for Parallel Kinematic Machine (PKM) are also discussed in this paper.

Functional Equations and Fourier Analysis

By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation-on compact groups.