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- Trial solution (2)
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- Additive manufacturing Laser beam trajectory optimization Powder bed fusion printing Heat simulation Linear-quadratic control (1)
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The fast classification of shapes is an important problem in shape analysis and of high relevance for many possible applications. In this paper, we consider the use of very fast and easy to compute statistical techniques for assessing shapes, which may for instance be useful for a first similarity search in a shape database. To this end, we construct shape signatures at hand of stochastic sampling of distances between points of interest in a given shape. By employing the Kolmogorov-Smirnov statistics we then propose to formulate the problem of shape classification as a statistical hypothesis test that enables to assess the similarity of the signature distributions. In order to illustrate some important properties of our approach, we explore the use of simple sampling techniques. At hand of experiments conducted with a variety of shapes in two dimensions, we give a discussion of potentially interesting features of the method.
n context of laser powder bed fusion (L-PBF), it is known that the properties of the final fabricated product highly depend on the temperature distribution and its gradient over the manufacturing plate. In this paper, we propose a novel means to predict the temperature gradient distributions during the printing process by making use of neural networks. This is realized by employing heat maps produced by an optimized printing protocol simulation and used for training a specifically tailored recurrent neural network in terms of a long short-term memory architecture. The aim of this is to avoid extreme and inhomogeneous temperature distribution that may occur across the plate in the course of the printing process. In order to train the neural network, we adopt a well-engineered simulation and unsupervised learning framework. To maintain a minimized average thermal gradient across the plate, a cost function is introduced as the core criteria, which is inspired and optimized by considering the well-known traveling salesman problem (TSP). As time evolves the unsupervised printing process governed by TSP produces a history of temperature heat maps that maintain minimized average thermal gradient. All in one, we propose an intelligent printing tool that provides control over the substantial printing process components for L-PBF, i.e. optimal nozzle trajectory deployment as well as online temperature prediction for controlling printing quality.
Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage
(2022)
Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications.
Feedforward neural networks offer a possible approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature. Computational approaches are in general highly dependent on a variety of computational parameters as well as on the choice of optimisation methods, a point that has to be seen together with the structure of the cost function. The intention of this paper is to make a step towards resolving these open issues. To this end, we study here the solution of a simple but fundamental stiff ordinary differential equation modelling a damped system. We consider two computational approaches for solving differential equations by neural forms. These are the classic but still actual method of trial solutions defining the cost function, and a recent direct construction of the cost function related to the trial solution method. Let us note that the settings we study can easily be applied more generally, including solution of partial differential equations. By a very detailed computational study, we show that it is possible to identify preferable choices to be made for parameters and methods. We also illuminate some interesting effects that are observable in the neural network simulations. Overall we extend the current literature in the field by showing what can be done in order to obtain useful and accurate results by the neural network approach. By doing this we illustrate the importance of a careful choice of the computational setup.
This paper presents a novel approach to distinguishing driving styles with respect to their energy efficiency. A distinct property of our method is that it relies exclusively on the global positioning system (GPS) logs of drivers. This setting is highly relevant in practice as these data can easily be acquired. Relying on positional data alone means that all features derived from them will be correlated, so we strive to find a single quantity that allows us to perform the driving style analysis. To this end we consider a robust variation of the so-called "jerk" of a movement. We give a detailed analysis that shows how the feature relates to a useful model of energy consumption when driving cars. We show that our feature of choice outperforms other more commonly used jerk-based formulations for automated processing. Furthermore, we discuss the handling of noisy, inconsistent, and incomplete data, as this is a notorious problem when dealing with real-world GPS logs. Our solving strategy relies on an agglomerative hierarchical clustering combined with an L-term heuristic to determine the relevant number of clusters. It can easily be implemented and delivers a quick performance, even on very large, real-world datasets. We analyse the clustering procedure, making use of established quality criteria. Experiments show that our approach is robust against noise and able to discern different driving styles
Computational Analysis of PDE-Based Shape Analysis Models by Exploring the Damped Wave Equation
(2022)
The computation of correspondences between shapes is a principal task in shape analysis. In this work, we consider correspondences constructed by a numerical solution of partial differential equations (PDEs). The underlying model of interest is thereby the classic wave equation, since this may give the most accurate shape matching. As has been observed in previous works, numerical time discretisation has a substantial influence on matching quality. Therefore, it is of interest to understand the underlying mechanisms and to investigate at the same time if there is an analytical model that could best describe the most suitable method for shape matching. To this end, we study here the damped wave equation, which mainly serves as a tool to understand and model properties of time discretisation. At the hand of a detailed study of possible parameters, we illustrate that the method that gives the most reasonable feature descriptors benefits from a damping mechanism which can be introduced numerically or within the PDE. This sheds light on some basic mechanisms of underlying computational and analytic models, as one may conjecture by our investigation that an ideal model could be composed of a transport mechanism and a diffusive component that helps to counter grid effects.
Collocation polynomial neural forms and domain fragmentation for solving initial value problems
(2022)
Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance, one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article, we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first-order polynomials. In this work, we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability.
A Novel Approach for Computation of Morphological Operations Using the Number Theoretic Transform
(2022)
The fundamental operations of mathematical morphology are dilation and erosion. They are often implemented using a sliding window with the purpose to compute maximum respectively minimum of pixel values within the corresponding mask.
We reformulate the problem of morphological dilation respectively erosion of an image with a non-flat filter as a convolution of their umbras. To this end, we propose to make use of the number theoretic transform to compute the convolution in this setting. In contrast to other possible schemes, this transform represents a completely discrete computational approach. It allows exact convolution of sequences made up of integers. Therefore we propose by the combination of umbra framework and number theoretic transform a well-engineered combination.
There is no restriction on size or shape of the structuring element, and also flat and non-flat filters can be realised.
An Exact Fast Fourier Method for Morphological Dilation and Erosion Using the Umbra Technique
(2022)
In this paper we consider the fundamental operations dilation and erosion of mathematical morphology. It is well known that many powerful image filtering operations can be constructed by their combinations. We propose a fast and novel algorithm based on the Fast Fourier Transform to compute grey-value morphological operations on an image. The novel method may deal with non-flat filters and incorporates no restrictions on shape and size of the filtering window, in contrast to many other fast methods in the field. Unlike fast Fourier techniques from previous works, the novel method gives exact results and is not an approximation. The key aspect which allows to achieve this is to explore here for the first time in this context the umbra formulation of images and filters. We show that the new method is in practice particularly suitable for filtering images with small tonal range or when employing large filter sizes.
The basic filters in mathematical morphology are dilation and erosion. They are defined by a flat or non-flat structuring element that is usually shifted pixel-wise over an image and a comparison process that takes place within the corresponding mask. The algorithmic complexity of fast algorithms that realise dilation and erosion for color images usually depends on size and shape of the structuring element.
In this paper we propose and investigate an easy and fast way to make use of the fast Fourier transform for an approximate computation of dilation and erosion for color images. Similarly in construction as many other fast algorithms, the method extends a recent scheme proposed for single-channel filtering. It is by design highly flexible, as it can be used with flat and non-flat structuring elements of any size and shape. Moreover, its complexity only depends on the number of pixels in the filtered images. We analyse here some important aspects of the approximation, and we show experimentally that we obtain results of very reasonable quality while the method has very attractive computational properties.