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The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models.
Scaling (or non-dimensionalization) is a mathematical te...
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Resumen Un colector lineal solar tipo Fresnel utiliza la energía solar para elevar la temperatura de un fluido mientras recorre un tubo colocado en el punto focal de un reflector compuesto por espejos planos. En el siguiente trabajo se realizará la simulación multifísica del tubo absorbedor, en donde ocurre el fenómeno de transferencia de calor y m...
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... Second, the ϵ-covergence criterion in (2.2) depends on the scale of the data for every system of interest, while the normalization allows to compare the nnGParareal performance across different ODE/PDE systems. Normalization is simple to apply in the context of ODEs/PDEs, as it reduces to a change of variables (see [31], Section 2.1.3 for more details). ...
With the advent of supercomputers, multi-processor environments and parallel-in-time (PinT) algorithms offer ways to solve initial value problems for ordinary and partial differential equations (ODEs and PDEs) over long time intervals, a task often unfeasible with sequential solvers within realistic time frames. A recent approach, GParareal, combines Gaussian Processes with traditional PinT methodology (Parareal) to achieve faster parallel speed-ups. The method is known to outperform Parareal for low-dimensional ODEs and a limited number of computer cores. Here, we present Nearest Neighbors GParareal (nnGParareal), a novel data-enriched PinT integration algorithm. nnGParareal builds upon GParareal by improving its scalability properties for higher-dimensional systems and increased processor count. Through data reduction, the model complexity is reduced from cubic to log-linear in the sample size, yielding a fast and automated procedure to integrate initial value problems over long time intervals. First, we provide both an upper bound for the error and theoretical details on the speed-up benefits. Then, we empirically illustrate the superior performance of nnGParareal, compared to GParareal and Parareal, on nine different systems with unique features (e.g., stiff, chaotic, high-dimensional, or challenging-to-learn systems).
... Non dimensionalization has been performed in prior modeling work on both LaMSA and MDA systems (Galantis and Woledge 2003;Ilton et al. 2018;Labonte 2023). For nondimensionalizing the dynamics of a mechanical system, we needed to choose three characteristic quantities: a characteristic mass , a characteristic length , and a characteristic time (for a survey of the principles of nondimensionalization, see Langtangen and Pedersen 2016). Here we constructed a characteristic length, mass, and time based on the properties of the apodeme because the role of the apodeme spring is central in this work. ...
Understanding the relationship between morphology and movement in biomechanical systems, particularly those composed of multiple complex elements, presents challenges due to the nonlinear nature of the interaction between components. This study focuses on the mandibular closing mechanisms in ants, specifically comparing muscle-driven actuation (MDA) and latch-mediated spring actuation (LaMSA) in the genus Strumigenys . Analyzing 3D structural data from diverse Strumigenys species, we employ mathematical models for both LaMSA and MDA systems. Our findings reveal distinct patterns of mechanical sensitivity between the two models, with sensitivity varying across kinematic output metrics. We explore the performance transition between MDA and LaMSA systems by incorporating biological data and correlations between morphological parameters into the models. In these models tuned specifically to Strumigenys , we find the LaMSA mechanism outperforms MDA at small relative mandible mass. Notably, the location and abruptness of the performance transition differs among various kinematic performance metrics. Overall, this work contributes a novel approach to understanding form-function relationships in complex biomechanical systems. By using morphological data to calibrate a general biomechanical model for a particular group, it strikes a balance between simplicity and specificity and allows for conclusions that are uniquely tuned to the morphological characteristics of the group.
... Since the set of variables (ψ, p, n) as well as physical constants exhibit values difference of many orders of magnitude, it is advantageous to scale the equations [5] and geometric domain so that the problem becomes dimensionless. The values can be easily scaled back to physical dimensions after the solution by multiplying the dimensionless value by its scaling constant according to: ...
... Units are essential when we deal with approximations, uncertainties and errors (Langtangen & Pedersen, 2016). An example is, a trait measurement where the length scale is typically measured in mm and approximates 12.5 m to the exact value of 12.52 m with an error of 0.02 m. ...
... A study working in mm would report 2 x10 2 as the error, while a study working in m would report 0.02 as the error. As a result, knowing the original measurement units is essential, and the downstream use of the unitless relative error is recommended (Langtangen & Pedersen, 2016). ...
Traits have become a crucial part of ecological and evolutionary sciences, helping researchers understand the function of an organism's morphology, physiology, growth and life history, with effects on fitness, behaviour, interactions with the environment and ecosystem processes. However, measuring, compiling and analysing trait data comes with data-scientific challenges.
We offer 10 (mostly) simple rules, with some detailed extensions, as a guide in making critical decisions that consider the entire life cycle of trait data.
This article is particularly motivated by its last rule, that is, to propagate good practice. It has the intention of bringing awareness of how data on the traits of organisms can be collected and managed for reuse by the research community.
Trait observations are relevant to a broad interdisciplinary community of field biologists, synthesis ecologists, evolutionary biologists, computer scientists and database managers. We hope these basic guidelines can be useful as a starter for active communication in disseminating such integrative knowledge and in how to make trait data future-proof. We invite the scientific community to participate in this effort at http://opentraits.org/best-practices.html.
... Astrophysics and cosmology Dolan 26 Kurth 30 Wesson 25 Chemical engineering Dobre and Marcano 31 Herschbach et al. 32 Worstell 33 Zlokarnik 34 Control engineering Balaguer 35 Brennan 36 Differential equations Alhama et al. 37 Petter Langtangen and Pedersen 38 Sánchez Pérez et al. 39 Experimental design, similitude, and modeling Albrecht et al. 40 Kline 41 Kuneš 42 Samarskii and Mikhailov 43 Szücs 44 Westine et al. 45 Fluid mechanics Islam 46 Sposito 47 Yarin 48 Food processes Delaplace et al. 49 Historical Macagno 5 Mathematical foundations Atherton 27 Curtis et al. 50 Siano 7,8 ...
The model based systems engineering (MBSE) approach describes a system using consistent views to provide a holistic model as complete as possible. MBSE methodologies end with the physical architecture of the system, but a physical model is clearly incomplete without the study of its associated physical laws and phenomena related to the whole system or its parts. However, the computational demands could be excessive even for modest projects. Dimensional analysis (DA) is common in fluid dynamics and chemical engineering, but its application to systems engineering is still limited. We describe an engineering methodological process, which incorporates DA as a powerful tool to understand the physical constraints of the system without the burden of complex analytical or numerical calculations. A detailed example describing a microantenna is presented showing the benefits of this approach. The selected example describes a problem rarely covered in modern expositions of DA in order to show the wide benefit of these techniques. The information provided by this analysis is very useful to select the best physically realizable architectures, testing design, and conduct trade-off studies. The complexity of modern systems and systems of systems demands new testing procedures in order to comply with increasingly demanding requirements and regulations. This can be accomplished through research in new DA methods. Finally, this article serves as a fairly comprehensive guide to the use of DA in the context of MBSE, detailing its strengths, limitations, and controversial issues
... Furthermore, the PINN training is a multi-objective optimization problem that requires a balanced form of all loss function terms. A complete procedure to obtain a dimensionless system with scaling techniques applied to differential equations is available in Langtangen and Pedersen (2016) . ...
A virtual flow meter (VFM) is an interesting and growing method to replace or support expensive and high-maintenance dependent physical flow rate sensors in oil and gas production systems. This paper presents a new use of Physics-Informed Neural Networks (PINN) as a hybrid virtual sensor to be used in an oil well system. This strategy combines prior knowledge about the system dynamics in the form of a phenomenological model in order to drive a long short-term memory (LSTM)-type recurrent neural network (RNN) training. The resulting model is able to predict the average flow rate in the production column some time steps ahead using measurements of states and exogenous inputs available in an oil well information system. The proposed scheme is also flexible as it can estimate fluid properties via their model parameters as an additional feature. The results obtained in this study through simulation used a benchmark electrical submersible pump (ESP) model to demonstrate the proposed PINN-based VFM potential as an accurate alternative to infer flow rate in oil wells.
... As mentioned in [71], models in solid mechanics are almost never non-dimensionalized, but our subsequent results in Section 3.1 will show its effectiveness. We introduce the following dimensionless variables with scaling factors u c and l c : ...
Recently, a class of machine learning methods called physics-informed neural networks (PINNs) has been proposed and gained great prevalence in solving various scientific computing problems. This approach enables the solution of partial differential equations (PDEs) via embedding physical laws into the loss function of neural networks. Many inverse problems can also be tackled by simply combining the data from real life scenarios with existing PINN algorithms. In this paper, we present a multi-task learning method to improve the training stability of PINNs for linear elastic inverse problems, and the homoscedastic uncertainty is introduced as a basis for weighting losses. Furthermore, we demonstrate an application of PINNs to a practical inverse problem in structural analysis: prediction of external loads of diverse engineering structures based on a limited number of displacement monitoring points. To this end, we first determine a simplified loading scenario at the offline stage. By setting unknown boundary conditions as learnable parameters, PINNs can predict the external loads applied to the structures with the support of enough measurement data. When it comes to the online stage in real engineering projects, transfer learning is employed to fine-tune the pre-trained model from offline stage. Our results show that, even with noisy gappy data, satisfactory results can still be obtained from the PINN model due to the dual regularization of physics laws and prior knowledge, which exhibits better robustness compared to traditional analysis methods. Our approach is capable of bridging the gap between various structures with geometric scaling and under different loading scenarios, and the convergence of training is also accelerated, thus making it possible for PINNs to be applied as surrogate models in actual engineering projects.
... Section 14. For inspiration, consider the famous Buckingham Theorem, which is based upon unit compatibility, and is used (particularly within the aerospace community) to derive physical laws centered around nondimensional numbers (e.g., Reynolds The methodology employed here is described more fully by [140]. Another consequence of the scaling in distance is that the terminal condition is, ...
Supremacy in armed conflict comes not merely from superiority in capability or numbers but from how assets are used, down to the maneuvers of individual vehicles and munitions. This document outlines a research plan focused on skirmish-level tactics to militarily relevant scenarios. Skirmish-level refers to both the size of the adversarial engagement -- generally one vs. one, two vs. one, and/or one vs. two -- as well as the fact that the goal or objective of each team is well-established. The problem areas include pursuit-evasion and target guarding, either of which may be considered as sub-problems within military missions such as air-to-air combat, suppression/defense of ground-based assets, etc. In most cases, the tactics considered are comprised of the control policy of the agents (i.e., their spatial maneuvers), but may also include role assignment (e.g, whether to act as a decoy or striker) as well as discrete decisions (e.g., whether to engage or retreat). Skirmish-level tactics are important because they can provide insight into how to approach larger scale conflicts (many vs. many, many objectives, many decisions). Machine learning approaches such as reinforcement learning and neural networks have been demonstrated to be capable of developing controllers for large teams of agents. However, the performance of these controllers compared to the optimal (or equilibrium) policies is generally unknown. Differential Game Theory provides the means to obtain a rigorous solution to relevant scenarios in the form of saddle-point equilibrium control policies and the min/max (or max/min) cost / reward in the case of zero-sum games. When the equilibrium control policies can be obtained analytically, they are suitable for onboard / real-time implementation. Some challenges associated with the classical Differential Game Theory approach are explored herein. These challenges arise mainly due to the presence of singularities, which may appear in even the simplest differential games. The utility of skirmish-level solutions is demonstrated in (i) the multiple pursuer, single evader differential games, (ii) multi-agent turret defense scenarios, and (iii) engage or retreat scenarios. In its culmination, this work contributes differential game and optimal control solutions to novel scenarios, numerical techniques for computing singular surfaces, approximations for computationally-intensive solutions, and techniques for addressing scenarios with multiple stages or outcomes.
... Mientras que, las unidades de los parámetros con * se proponen debido a la falta de explicación en el artículo [75], estos se obtienen de sustituir las unidades de las variables t, M y B y los parámetros en el modelo (2.4) que se tienen, de tal forma que se satisfagan las igualdades del sistema. Para mas detalles adimensionalización de EDO's, véase [69], Capítulo 2. ...
n the present work, systems of differential equations are studied and proposed,
that describe the dynamics of hemiparasitic plants, their hosts and to
In the case of mistletoe, its vectors (birds), which have the characteristic
common to be multistable, that is, they have at least two attractors (punctual or
newspaper). Since they have different attractors, it is shown that it is possible to do
a transition from one region of attraction to another, under control or disturbance.
... As discussed in [15,16], the concentrations in thermodynamic equilibrium fulfill the following mass conservation laws ...
... To complete the mathematical system describing the thermodynamic equilibrium, as in [15], the definition of the complex-formation constant 0 < β κ of the κ-th reaction in (1), see [16], is used: ...
The limiting current density is one of to the most important indicators in electroplating for the maximal current density from which a metal can be deposited effectively from an electrolyte. Hence, it is an indicator of the maximal deposition speed and the homogeneity of the thickness of the deposited metal layer. For these reasons, a major interest in the limiting current density is given in practical applications. Usually, the limiting current density is determined via measurements. In this article, a simple model to compute the limiting current density is presented, basing on a system of diffusion–reaction equations in one spatial dimension. Although the model formulations need many assumptions, it is of special interest for screenings, as well as for comparative work, and could easily be adjusted to measurements.
