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Recent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here.
Numerical Analysis is a broad field, and coming to grips with all of it may seem like a daunting task. This text provides a thorough and comprehensive exposition of all the topics contained in a classical graduate sequence in numerical analysis. With an emphasis on theory and connections with linear algebra and analysis, the book shows all the rigor of numerical analysis. Its high level and exhaustive coverage will prepare students for research in the field and become a valuable reference as they continue their career. Students will appreciate the simple notation, clear assumptions and arguments, as well as the many examples and classroom-tested exercises ranging from simple verification to qualifying exam-level problems. In addition to the many examples with hand calculations, readers will also be able to translate theory into practical computational codes by running sample MATLAB codes as they try out new concepts.
The present analytical design of shrink fits typically results in an uneven stress condition that can lead to failure in a variety of manners. With increasing loads and the use of brittle materials, the optimization of the stresses in the shrink fit hence becomes increasingly necessary. Currently existing approaches do not solve the problem satisfactorily or increase the manufacturing and design effort. This paper therefore considers the implementation of an AI-based stress optimization using reinforcement learning, which performs stress optimization by geometrically contouring the interstice.
Increasing product complexity and individual customer requirements make the design of optimal product families difficult. Numerical optimization supports optimal design but must deal with the following challenges: many design variables, non-linear or discrete dependencies, and many possibilities of assigning shared components to products. Existing approaches use simplifications to alleviate those challenges. However, for use in industrial practice, they often use irrelevant commonality metrics, do not rely on the actual design variables of the product, or are unable to treat discrete variables. We present a two-level approach: (1) a genetic algorithm (GA) to find the best commonality scheme (i.e., assignment scheme of shared components to products) and (2) a particle swarm optimization (PSO) to optimize the design variables for one specific commonality scheme. It measures total cost, comprising manufacturing costs, economies of scales and complexity costs. The approach was applied to a product family consisting of five water hose boxes, each of them being subject to individual technical requirements. The results are discussed in the context of the product family design process.
One of the fundamental requirements for dual purpose casks, which are used for the transport and interim storage of spent fuel assemblies, is the safe removal of the resulting decay heat. To ensure this the temperature fields are determined using numerical methods. However, their modelling is complex and the computation time-consuming.
In order to accelerate this thermal assessment, we have developed z88ENSI, an independent simulation tool based on finite element analysis. With regard to the modelling, various parameters can be varied quickly with our newly designed mesh manipulation procedure. Concerning the computation time, we developed and implemented an approach for calculating three-dimensional temperature fields, based on an already existing two-dimensional method which lacked precision. We accelerate the calculation by using extended thermal gap constraints, which depict the thermal behaviour of the non-meshed, gas-filled gaps inside the cask. We validate the results of our calculation tool by comparing them with those generated with Ansys. The results of the comparison temperatures differ between −0.8% and 3.7%. The speedup of z88ENSI for the specific validation setting is between 6.9 and 15.0.
Political districts may be drawn to favor one group or political party over another, or gerrymandered. A number of measurements have been suggested as ways to detect and prevent such behavior. These measures give concrete axes along which districts and districting plans can be compared. However, measurement values are affected by both noise and the compounding effects of seemingly innocuous implementation decisions. Such issues will arise for any measure. As a case study demonstrating the effect, we show that commonly used measures of geometric compactness for district boundaries are affected by several factors irrelevant to fairness or compliance with civil rights law. We further show that an adversary could manipulate measurements to affect the assessment of a given plan. This instability complicates using these measurements as legislative or judicial standards to counteract unfair redistricting practices. This paper accompanies the release of packages in C++, Python, and R that correctly, efficiently, and reproducibly calculate a variety of compactness scores.
In the domain of optical engineering, optomechatronic systems are predominantly developed using conventional ray tracing methods such as sequential and non-sequential ray tracing. However, the increasing complexity of these systems in combination with the demand for high efficiency and high image quality leads to the fact that conventional methods to develop these systems reach their limits. In order to be able to develop highly efficient systems with high image quality, this contribution introduces a hybrid ray tracing method using an advanced optimization function.
The observational properties of isolated NSs are shaped by their magnetic field and surface temperature. They evolve in a strongly coupled fashion, and modelling them is key in understanding the emission properties of NSs. Much effort was put in tackling this problem in the past but only recently a suitable 3D numerical framework was developed. We present a set of 3D simulations addressing both the long-term evolution (≈ 104–106 yrs) and short-lived outbursts (≲ 1 yr). Not only a 3D approach allows one to test complex field geometries, but it is absolutely key to model magnetar outbursts, which observations associate to the appearance of small, inherently asymmetric hot regions. Even though the mechanism that triggers these phenomena is not completely understood, following the evolution of a localised heat injection in the crust serves as a model to study the unfolding of the event.
Anchored in simple and familiar physics problems, the author provides a focused introduction to mathematical methods in a narrative driven and structured manner. Ordinary and partial differential equation solving, linear algebra, vector calculus, complex variables and numerical methods are all introduced and bear relevance to a wide range of physical problems. Expanded and novel applications of these methods highlight their utility in less familiar areas, and advertise those areas that will become more important as students continue. This highlights both the utility of each method in progressing with problems of increasing complexity while also allowing students to see how a simplified problem becomes 're-complexified'. Advanced topics include nonlinear partial differential equations, and relativistic and quantum mechanical variants of problems like the harmonic oscillator. Physics, mathematics and engineering students will find 300 problems treated in a sophisticated manner. The insights emerging from Franklin's treatment make it a valuable teaching resource.
Recently, design researchers have begun to use neuroimaging methods (e.g., functional magnetic resonance imaging, fMRI) to understand a variety of cognitive processes relevant to design. However, common neuroimaging analysis techniques require significant assumptions relating temporal and spatial information during model formulation. In this work, we apply hidden Markov Models (HMM) in order to uncover patterns of brain activation in a design-relevant fMRI dataset. The underlying fMRI data comes from a prior research study in which participants generated solutions for twelve open-ended design problems from the literature. HMMs are generative models that are able to automatically infer the internal state characteristics of a process by observing state emissions. In this work, we demonstrate that distinct states can be extracted from the design ideation fMRI dataset, and that designers are likely to transition between a few key states. Additionally, the likelihood of occupancy within these states is different for high and low performing designers. This work opens up the door for future research to investigate the patterns of neural activation within the discovered states.
Gears are essential machine elements in the drivetrain and transmission technology. The operational behaviour of a gear pairing is influenced by the design of the gear kinematics as well as the component properties. With regard to an improvement of performance and service life, the targeted modification of tooth geometry and component properties offers a promising approach. Thus, the achievable geometric and mechanical component properties are influenced by the manufacturing process, which must be taken into account in the design process. The application of virtual evaluation methods is suitable for this purpose. For the manufacturing of steel gears, cold forging provides the potential of achieving beneficial mechanical properties in a highly productive process. Major challenges for the industrial application are the short service life of the cost- intensive tools and the low geometric accuracy in comparison to machining processes. Within this study the design of the tooth geometry as well as the associated forming tool are investigated. The aim is to derive recommendations regarding an optimization of the resulting component properties and operational behaviour.
Basically, the safe dissipation of heat is among others an important protection objective of dual purpose casks. Gas-filled gaps within such casks can play a major role for the thermal behavior as they act as thermal barriers due to the lower heat conductivity of gaseous fluids in comparison to metallic materials. However, additional heat transmission mechanisms, such as natural convection and radiation can also occur in a gaseous medium. This leads to both an expanded modelling and a prolonged computing time in numerical simulations. Within the scope of a research project in cooperation with Swiss Federal Nuclear Safety Inspectorate ENSI a simulation tool for the fast thermal evaluation of dual purpose casks is developed which combines analytical methods and FEA. The innovation is that the thermal effects of gas-filled gaps are considered by using analytical equations. Main focus lies on the implementation of heat radiation as a non-linear transfer mechanism. Therefore, an iterative calculation process is used and the effects of the iteration number is investigated. Furthermore, the influence of radiation in comparison to pure conduction is examined depending on the gap width.
As self-gravitating systems, dense star clusters exhibit a natural diffusion of energy from their innermost to outermost regions, leading to a slow and steady contraction of the core until it ultimately collapses under gravity. However, in spite of the natural tendency toward “core collapse,” the globular clusters (GCs) in the Milky Way exhibit a well-observed bimodal distribution in core radii separating the core-collapsed and non-core-collapsed clusters. This suggests an internal energy source is at work, delaying the onset of core collapse in many clusters. Over the past decade, a large amount of work has suggested that stellar black holes (BHs) play a dynamically-significant role in clusters throughout their entire lifetimes. Here we review our latest understanding of BH populations in GCs and demonstrate that, through their dynamical interaction with their host cluster, BHs can naturally explain the distinction between core-collapsed and non-core-collapsed clusters through a process we call “black hole burning.”
Chapter 11 introduces the reader to the world of direct numerical simulations. Temporal and spatial formulations are covered along with boundary and initial conditions. Time-marching methods and spatial discretization methods are also discussed. A variety of applications are then presented.
Given a raw data sample, the purpose of this paper is to design a numerical procedure to model this sample under the form of polynomial chaos expansion. The coefficients of the polynomial are computed as the solution to a constrained optimization problem. The procedure is first validated on samples coming from a known distribution and it is then applied to raw experimental data of unknown distribution. Numerical experiments show that only five coefficients of the Chaos expansions are required to get an accurate representation of a sample.
This paper provides conditions under which an algorithm for the computation and simulation of Bewley–Huggett–Aiyagari models, based on state-space discretization, will converge to all true solutions. These conditions are shown to be satisfied in two standard examples: the Aiyagari model and its extension to endogenous labor. Bewley–Huggett– Aiyagari models are general equilibrium models with incomplete markets and idiosyncratic, but no aggregate, shocks. The algorithm itself is based on discretization, while the theory importantly allows for making simulations using the approximate computational solution of the value function problem rather than the true model solution. The numerical results of applying the algorithm to both models are provided and investigated in terms of replication, revealing that the Aiyagari model overestimates the degree of precautionary savings in the high-risk-and-high-risk-aversion case. The results also show that both models almost certainly have a unique general equilibrium. Theoretically, the existence of equilibria was known, but uniqueness remained an open question.
In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn–Hilliard-Reaction model with elasticity in the framework of the Cahn–Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in lithium ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface intersects the free boundary of the Si layer. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.
It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the “pollution error” caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weakform. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.
We have performed 3D hydrodynamic simulations of a symmetrical jet ejection following previous works (Raga et al. 2009, Riera et al. 2014, Velázquez et al. 2014). The jet is emitted from a binary system in elliptical orbit, and its direction changes describing a precession cone. We have considered that the jet has a time-dependence density ejection or a time-dependence velocity ejection, in order to propose an alternative model to explain the morphology of PPNe’s. Also in our description we have included the effect of the photoionization of the central source. From numerical results, synthetic Hα maps were obtained, and a proper motion study were carried out. We found that the photoionization has an important effect on the case with variation density resulting in a increse in the Hα emission.