Conservation of energy

Summary

In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. Energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance, chemical energy is converted to kinetic energy when a stick of dynamite explodes. If one adds up all forms of energy that were released in the explosion, such as the kinetic energy and potential energy of the pieces, as well as heat and sound, one will get the exact decrease of chemical energy in the combustion of the dynamite. Classically, conservation of energy was distinct from conservation of mass. However, special relativity shows that mass is related to energy and vice versa by E = mc^2, the equation representing mass–energy equivalence, and science now takes the view that mass-energy as a whole is conserved. Theoretically, this implies that any object with

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Controlling oscillations in high-order schemes using neural networks

Niccolo' Discacciati

High-order numerical solvers for conservation laws suffer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce their amplitude aims to add a suitable amount of artificial viscosity. Several models are available in the literature, which rely on the identification of a shock sensor, adding dissipation when the solution regularity is lost. The dependence on problem-specific parameters limits their performances. To solve this issue, in this thesis we propose a new technique based on artificial neural networks. In particular, we focus on the construction of multilayer perceptrons. Emphasis is given to the training phase, which is carried out using a robust dataset created using the classical models with optimal parameters. The online evaluation is then integrated in the numerical solver of the partial differential equation. Even though most of the effort is put in one-dimensional problems, an extension to a two-dimensional scenario is provided. Several numerical results are presented to demonstrate the capabilities of the network-based technique. Initially, we focus on one-dimensional scalar problems, where Burgers equation represents our first benchmark test. Then, we move to less simple cases, characterized by non-convex flux functions or involving multiple equations (compressible Euler equations). The same strategy is followed for multi-dimensional problems. In most of the cases, the proposed model is able to guarantee high accuracy in presence of smooth solutions and to capture discontinuities (shock and contact waves). In general, the results are comparable to (or better than) the classical models with properly tuned parameters. A final performance analysis is carried out.

2018

Controlling oscillations in high-order Discontinuous Galerkin schemes using artificial viscosity tuned by neural networks

Niccolo' Discacciati, Jan Sickmann Hesthaven, Deep Ray

ACADEMIC PRESS INC ELSEVIER SCIENCE2020

Molten Salt Reactors is a class of advanced reactors that is typically characterised by the presence of a molten inorganic salt of a fissile material as fuel. At the moment, this class of reactors is considered one of the promising options in the long run of nuclear power.In the context of safety assessment of the MSFR, a task on analysis of selected transients was planned by the SAMOFAR project, to which this thesis should contribute. In addition, fluids with high melting temperature bring with them challenges regarding solidification of the fluid and possible flow blockage, to which a method was implemented during this thesis to evaluate the consequences. The open core cavity with curved shape adds another issue where the core is highly turbulent and requires the use of unstructured mesh codes for routine analysis, adding uncertainty and computational burden to the analytical workflow. Using ATARI, the tool developed during this thesis, we:1. Analyse the MSFR together with partners in support of a safety assessment task.2. Consider the impact and how to include solidification/melting in an analytical methodology.3. Evaluate options to manage/reduce uncertainties and the burden of routine use of high-fidelity codes.After a careful derivation of the conservation laws and assembling the code main algorithms, ATARI undergoes verification in a limited scope using the Stefan problem, manufactured solutions and energy balances.The nuclear cavity benchmark was performed between SAMOFAR partners showing equivalency of the codes within the scope of the benchmark. Then the participating institutions perform analysis of the MSFR for the safety assessment task, which ATARI is unable to followup with the transient analysis. The reason for failure is identified as the requirement of appropriate acceleration schemes by OpenFOAM-based multiphysics solvers, and the lack of one in ATARI.In the lack of accurate data regarding the heat exchangers, the impact of solidification is evaluated in a few simplified cases of a pipe with square cross-section and heat exchanger. For different salt compositions and different boundary conditions, the heat exchanger is found to recover from blockage if pump maintains its momentum and some cross-flow is allowed. The recovery of cases with abrupt solidification is questionable due to the required pressure head.The use of flow baffles is investigated as an option to structure the flow inside the reactor and reduce uncertainties originating from turbulence modelling while allowing the use of structured mesh codes for analysis. The concept is tested in a chloride based MSR, which retains its breed and burn mode while showing a quasi-1D flow appropriate for modelling with legacy codes.We recommend that the nuclear cavity benchmark should be expanded to include a more demanding transient case, representative of MSFR transient analysis requirements. A case is made for homogenised and economical models as used in ATARI to enable the practical use of coupling methods that promote the simultaneous solution of equations. A niche for ATARI is found as a special-mission code for analysis of challenging cases with deforming structure and flow field that justifies the use of unstructured meshes and the added computational burden. Due to design uncertainties and their nature, it is recommended to consider solidification implicitly when analysing the MSFR fuel circuit.

EPFL2021