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Elastic transmission eigenvalues and their computation via the method of fundamental solutions
(2020)
A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach.
Electric flight has the potential for a more sustainable and energy-saving way of aviation compared to fossil fuel aviation. The electric motor can be used as a generator inflight to regenerate energy during descent. Three different approaches to regenerating with electric propeller powertrains are proposed in this paper. The powertrain is to be set up in a wind tunnel to determine the propeller efficiency in both working modes as well as the noise emissions. Furthermore, the planned flight tests are discussed. In preparation for these tests, a yaw stability analysis is performed with the result that the aeroplane is controllable during flight and in the most critical failure case. The paper shows the potential for inflight regeneration and addresses the research gaps in the dual role of electric powertrains for propulsion and regeneration of general aviation aircraft.
This paper discusses a new way of inflight power regeneration for electric or hybrid-electric driven general aviation aircraft with one powertrain for both configurations. Three different approaches for the shift from propulsion to regeneration mode are analyzed. Numerical cal-culation and wind tunnel results are compared and show the highest regeneration potential for the "Windmill" approach, where the propeller blades are flipped, and rotation is reversed. A combination of all regeneration approaches for a realistic flight mission is discussed.
The development and operation of hybrid or purely electrically powered aircraft in regional air mobility is a significant challenge for the entire aviation sector. This technology is expected to lead to substantial advances in flight performance, energy efficiency, reliability, safety, noise reduction, and exhaust emissions. Nevertheless, any consumed energy results in heat or carbon dioxide emissions and limited electric energy storage capabilities suppress commercial use. Therefore, the significant challenges to achieving eco-efficient aviation are increased aircraft efficiency, the development of new energy storage technologies, and the optimization of flight operations. Two major approaches for higher eco-efficiency are identified: The first one, is to take horizontal and vertical atmospheric motion phenomena into account. Where, in particular, atmospheric waves hold exciting potential. The second one is the use of the regeneration ability of electric aircraft. The fusion of both strategies is expected to improve efficiency. The objective is to reduce energy consumption during flight while not neglecting commercial usability and convenient flight characteristics. Therefore, an optimized control problem based on a general aviation class aircraft has to be developed and validated by flight experiments. The formulated approach enables a development of detailed knowledge of the potential and limitations of optimizing flight missions, considering the capability of regeneration and atmospheric influences to increase efficiency and range.
Reducing poverty, protecting the planet, and improving life on earth for everyone are the essential goals of the "2030 Agenda for Sustainable Development"committed by the United Nations (UN). Achieving those goals will require technological innovation as well as their implementation in almost all areas of our business and day-to-day life. This paper proposes a high-level framework that collects and structures different uses cases addressing the goals defined by the UN. Hence, it contributes to the discussion by proposing technical innovations that can be used to achieve those goals. As an example, the goal "Climate Actionïs discussed in detail by describing use cases related to tackling biodiversity loss in order to conservate ecosystems.
The management of knowledge in organizations considers both established long-term processes and cooperation in agile project teams. Since knowledge can be both tacit and explicit, its transfer from the individual to the organizational knowledge base poses a challenge in organizations. This challenge increases when the fluctuation of knowledge carriers is exceptionally high. Especially in large projects in which external consultants are involved, there is a risk that critical, company-relevant knowledge generated in the project will leave the company with the external knowledge carrier and thus be lost. In this paper, we show the advantages of an early warning system for knowledge management to avoid this loss. In particular, the potential of visual analytics in the context of knowledge management systems is presented and discussed. We present a project for the development of a business-critical software system and discuss the first implementations and results.
The low-pressure system Bernd involved extreme rainfalls in the Western part of Germany in July 2021,
resulting in major floods, severe damages and a tremendous number of casualties. Such extreme events
are rare and full flood protection can never be ensured with reasonable financial means. But still, this
event must be starting point to reconsider current design concepts. This article aims at sharing some
thoughts on potential hazards, the selection of return periods and remaining risk with the focus on Germany.
We present new numerical results for shape optimization problems of interior Neumann eigenvalues. This field is not well understood from a theoretical standpoint. The existence of shape maximizers is not proven beyond the first two eigenvalues, so we study the problem numerically. We describe a method to compute the eigenvalues for a given shape that combines the boundary element method with an algorithm for nonlinear eigenvalues. As numerical optimization requires many such evaluations, we put a focus on the efficiency of the method and the implemented routine. The method is well suited for parallelization. Using the resulting fast routines and a specialized parametrization of the shapes, we found improved maxima for several eigenvalues.
The method of fundamental solutions is applied to the approximate computation of interior transmission eigenvalues for a special class of inhomogeneous media in two dimensions. We give a short approximation analysis accompanied with numerical results that clearly prove practical convenience of our alternative approach.
Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations.
In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric 2x2 matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.
Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary
(2024)
In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.
We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a C² boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach.
Analysis and computation of the transmission eigenvalues with a conductive boundary condition
(2022)
We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions we study are: deriving Faber–Krahn type lower bounds, the discreteness and limiting behavior of the transmission eigenvalues as the conductivity tends to infinity for a sign changing contrast. We also provide a numerical study of a new boundary integral equation for computing the eigenvalues. Lastly, using the limiting behavior we will numerically estimate the refractive index from the eigenvalues provided the conductivity is sufficiently large but unknown.
Fields of asymmetric tensors play an important role in many applications such as medical imaging (diffusion tensor magnetic resonance imaging), physics, and civil engineering (for example Cauchy-Green-deformation tensor, strain tensor with local rotations, etc.). However, such asymmetric tensors are usually symmetrized and then further processed. Using this procedure results in a loss of information. A new method for the processing of asymmetric tensor fields is proposed restricting our attention to tensors of second-order given by a 2x2 array or matrix with real entries. This is achieved by a transformation resulting in Hermitian matrices that have an eigendecomposition similar to symmetric matrices. With this new idea numerical results for real-world data arising from a deformation of an object by external forces are given. It is shown that the asymmetric part indeed contains valuable information.
An alternative method is presented to numerically compute interior elastic transmission eigenvalues for various domains in two dimensions. This is achieved by discretizing the resulting system of boundary integral equations in combination with a nonlinear eigenvalue solver. Numerical results are given to show that this new approach can provide better results than the finite element method when dealing with general domains.
The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.
There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost.
Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different.