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Numerical stability of tangents and adjoints of implicit functions

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arXiv 2021年

作者： Naumann, Uwe Informatik 12: Software and Tools for Computational Engineering RWTH Aachen University Germany

We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally numerically stable. Tangents of systems of linear equations can become instable as well as both tangents and adjoints of systems of nonlinear equations, which extends to optima of convex unconstrained objectives. Sufficient conditions for numerical stability are derived. © 2021, CC BY.

关键词： Nonlinear equations

Reducing Spatial Discretization Error on Coarse CFD Simulations Using an OpenFOAM-Embedded Deep Learning Framework

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arXiv 2024年

作者： Gonzalez-Sieiro, Jesus Pardo, David Nava, Vincenzo Calo, Victor M. Towara, Markus BCAM - Basque Center for Applied Mathematics Bilbao Spain Leioa Spain Ikerbasque - Basque Foundation for Sciences Bilbao Spain Derio Spain School of Electrical Engineering Computing and Mathematical Sciences Curtin University Perth Australia Software and Tools for Computational Engineering RWTH Aachen University Aachen Germany

We propose a method for reducing the spatial discretization error of coarse computational fluid dynamics (CFD) problems by enhancing the quality of low-resolution simulations using deep learning. We feed the model with fine-grid data after projecting it to the coarse-grid discretization. We substitute the default differencing scheme for the convection term by a feed-forward neural network that interpolates velocities from cell centers to face values to produce velocities that approximate the down-sampled fine-grid data well. The deep learning framework incorporates the open-source CFD code OpenFOAM, resulting in an end-to-end differentiable model. We automatically differentiate the CFD physics using a discrete adjoint code version. We present a fast communication method between TensorFlow (Python) and OpenFOAM (c++) that accelerates the training process. We applied the model to the flow past a square cylinder problem, reducing the error from 120% to 25% in the velocity for simulations inside the training distribution compared to the traditional solver using an x8 coarser mesh. For simulations outside the training distribution, the error reduction in the velocities was about 50%. The training is affordable in terms of time and data samples since the architecture exploits the local features of the physics. © 2024, CC BY.

关键词： computational fluid dynamics

Scheduled Jacobian Chaining

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arXiv 2025年

作者： Märtens, Simon Naumann, Uwe Informatik 12: Software and Tools for Computational Engineering RWTH Aachen University Aachen52056 Germany

This paper addresses the efficient computation of Jacobian matrices for programs composed of sequential differentiable subprograms. By representing the overall Jacobian as a chain product of the Jacobians of these subprograms, we reduce the problem to optimizing the sequence of matrix multiplications, known as the Jacobian Matrix Chain Product problem. Solutions to this problem yield "optimal bracketings", which induce a precedence-constraint scheduling problem. We investigate the inherent parallelism in the solutions and develop a new dynamic programming algorithm as a heuristic that incorporates the scheduling. To assess its performance, we benchmark it against the global optimum, which is computed via a branch-and-bound algorithm. © 2025, CC BY.

关键词： Jacobian matrices

On the computational complexity of the chain rule of differential calculus

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arXiv 2021年

作者： Naumann, Uwe Informatik 12: Software and Tools for Computational Engineering RWTH Aachen University AachenD-52056 Germany

Many modern numerical methods in computational science and engineering rely on derivatives of mathematical models for the phenomena under investigation. The computation of these derivatives often represents the bottleneck in terms of overall runtime performance. First and higher derivative tensors need to be evaluated efficiently. The chain rule of differentiation is the fundamental prerequisite for computing accurate derivatives of composite functions which perform a potentially very large number of elemental function evaluations. Data flow dependences amongst the elemental functions give rise to a combinatorial optimization problem. We formulate Chain Rule Differentiation and we prove it to be NP-complete. Pointers to research on its approximate solution are *** Codes 26B05, 68Q17 © 2021, CC BY-NC-ND.

关键词： Differentiation (calculus)

Evaluating Energy Efficiency of Buildings using Artificial Neural Networks and K-means Clustering Techniques 3

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Evaluating Energy Efficiency of Buildings using Artificial N...

3rd International Conference on Computing, Mathematics and engineering Technologies, iCoMET 2020

作者： Nazir, Ahsan Wajahat, Ahsan Akhtar, Faheem Ullah, Faheem Qureshi, Sirajuddin Malik, Sher Afghan Shakeel, Anum Beijing University of Technology Beijing School of Software Engineering China RWTH Aachen University Software Tools for Computational Engineering Germany Sukkur IBA University Department of Computer Science Sukkur65200 Pakistan

ISBN: (纸本)9781728149707

The consumption of energy in buildings has risen abruptly over the last decades. Due to less energy-efficient buildings, most of the energy is being thrown in our surroundings thus making an adverse effect on our environment. In this paper, heating and cooling loads of private or non-commercial buildings are covered. By implementing the proposed technique, which is a blend of cluster analysis and Artificial Neural Network (ANN), evaluation and prediction are performed. The estimation of heating and cooling loads of private or non-commercial buildings are performed using eight input variables in the ANN-based model. The details of variables are as follows, a relative surface area, total height, compactness, roof area, glazing area distribution, orientation, glazing area, and wall area. K-means clustering methodology is then used to cluster buildings on the basis of output variables. Stand on simulated literature data, evaluation of 768 different private or non-commercial buildings is done using the above-suggested method. Research results depicted that depending upon input variables, the above-suggested approach can efficiently evaluate heating and cooling load that is very much close to real test results. © 2020 IEEE.

关键词： Cluster analysis

Optimization of generalized Jacobian chain products without memory constraints

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arXiv 2020年

作者： Naumann, Uwe Software and Tools for Computational Engineering RWTH Aachen University Germany

The efficient computation of Jacobians represents a fundamental challenge in computational science and engineering. Large-scale modular numerical simulation programs can be regarded as sequences of evaluations of in our case differentiable modules with corresponding local Jacobians. The latter are typically not available. Tangent and adjoint versions of the individual modules are assumed to be given as results of algorithmic differentiation instead. The classical (Jacobian) matrix chain product formulation is extended with the optional evaluation of matrix-free Jacobian-matrix and matrix-Jacobian products as tangents and adjoints. We propose a dynamic programming algorithm for the minimization of the computational cost of such generalized Jacobian chain products without considering constraints on the available persistent system memory. In other words, the naive evaluation of an adjoint of the entire simulation program is assumed to be a feasible option. No checkpointing is required. Under the given assumptions we obtain optimal solutions which improve the best state of the art methods by factors of up to seven on a set of randomly generated problem instances of growing size. Copyright © 2020, The Authors. All rights reserved.

关键词： Dynamic programming

Evaluating Energy Efficiency of Buildings using Artificial Neural Networks and K-means Clustering Techniques

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Evaluating Energy Efficiency of Buildings using Artificial N...

International Conference on Computing, Mathematics and engineering Technologies (iCoMET)

作者： Ahsan Nazir Ahsan Wajahat Faheem Akhtar Faheem Ullah Sirajuddin Qureshi Sher Afghan Malik Anum Shakeel School of Software Engineering Beijing University of Technology Beijing China Software & Tools for Computational Engineering RWTH Aachen University Germany Department of Computer Science Sukkur IBA University Sukkur Pakistan

ISBN: (数字)9781728149707

ISBN: (纸本)9781728149714

关键词： Buildings Heating systems Cooling Neural networks Data models Load modeling Predictive models

Physical reflective boundary conditions applied to smoothed particle hydrodynamics (SPH) method for solving fluid dynamics problems in 3-D domains 6

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Physical reflective boundary conditions applied to smoothed ...

6th International Conference on Particle-Based Methods. Fundamentals and Applications, PARTICLES 2019

作者： Fraga Filho, C.A.D. Peng, C. Islam, M.R.I. McCabe, C. Baig, S. Durga Prasad, G.V. ESS Engineering Software GmbH Berggasse 35 Steyr4400 Austria Development Implementation and Application of Computational Tools for Problem Solving in Engineering Research Group Federal Institute of Education Science and Technology of Espírito Santo Av. Vitória 1729 Jucutuquara Vitória–ES29040-780 Brazil

ISBN: (纸本)9788412110111

This paper aims to present the physical reflective boundary conditions (RBC) in 3-D domains and applications in fluid dynamics problems. Currently, RBC have been used in meshfree particle methods as an attempt to respect the continuum physical laws at the macroscopic domain, without the application of fictitious/ ghost particles - that improperly mix molecular and continuum mechanics concepts [1]. RBC methodology, validation in 2-D domains and applications in hydrostatics and hydrodynamics cases were presented in [2]. Hydrostatics and hydrodynamics cases with the RBC implementation in 3-D domains are presented in this work. A Newtonian, incompressible, uniform and isothermal fluid inside an immobile reservoir open to the atmosphere and dam break flow have been studied. In the first case, a modified SPH formulation using a modified pressure concept [3] has been used. Dam breaking simulations used the standard SPH formulation. In both cases, the numerical results showed good agreement with the analytical results or literature data. © 2019 The Authors.

关键词： Boundary conditions

Erratum: “Implementation of three-dimensional physical reflective boundary conditions in mesh-free particle methods for continuum fluid dynamics: Validation tests and case studies” [Phys. Fluids 31, 103606 (2019)]

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Physics of Fluids 2020年第2期32卷

作者： Carlos Alberto Dutra Fraga Filho Chong Peng Md Rushdie Ibne Islam Christopher McCabe Samiullah Baig Gonella Venkata Durga Prasad 1ESS Engineering Software Steyr GmbH Berggasse 35 4400 Steyr Austria 2Development Implementation and Application of Computational Tools for Problem Solving in Engineering Research Group Federal Institute of Education Science and Technology of Espírito Santo Avenida Vitória 1729 Jucutuquara 29040-780 Vitória-ES Brazil

In the original paper,1 on p. 2, in the Nomenclature section, “h support radius” should be “h smoothing length” and on p. 18 “A support radius that varied with

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