scholarly journals Provably non-stiff implementation of weak coupling conditions for hyperbolic problems

2022 ◽  
Author(s):  
Ossian O’Reilly ◽  
Jan Nordström
Keyword(s):  
Numerical Experiments ◽  
Numerical Scheme ◽  
General Procedure ◽  
Model Parameters ◽  
Hyperbolic Problems ◽  
Linear Functionals ◽  
Time Step ◽  
Coupling Conditions ◽  
Coupling Procedure ◽  
Projection Matrices

AbstractIn the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant–Friedrichs–Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals.

scholarly journals MHDSTS: a new explicit numerical scheme for simulations of partially ionised solar plasma

2018 ◽  
Vol 615 ◽  
pp. A67 ◽  
Author(s):  
P. A. González-Morales ◽  
E. Khomenko ◽  
T. P. Downes ◽  
A. de Vicente
Keyword(s):  
Numerical Scheme ◽  
Operator Splitting ◽  
Conservation Equation ◽  
Ambipolar Diffusion ◽  
Point Of View ◽  
Integration Time ◽  
Solar Photosphere ◽  
Time Step ◽  
Diffusion Term ◽  
Fluid Approximation

The interaction of plasma with magnetic field in the partially ionised solar atmosphere is frequently modelled via a single-fluid approximation, which is valid for the case of a strongly coupled collisional media, such as solar photosphere and low chromosphere. Under the single-fluid formalism the main non-ideal effects are described by a series of extra terms in the generalised induction equation and in the energy conservation equation. These effects are: Ohmic diffusion, ambipolar diffusion, the Hall effect, and the Biermann battery effect. From the point of view of the numerical solution of the single-fluid equations, when ambipolar diffusion or Hall effects dominate can introduce severe restrictions on the integration time step and can compromise the stability of the numerical scheme. In this paper we introduce two numerical schemes to overcome those limitations. The first of them is known as super time-stepping (STS) and it is designed to overcome the limitations imposed when the ambipolar diffusion term is dominant. The second scheme is called the Hall diffusion scheme (HDS) and it is used when the Hall term becomes dominant. These two numerical techniques can be used together by applying Strang operator splitting. This paper describes the implementation of the STS and HDS schemes in the single-fluid code MANCHA3D. The validation for each of these schemes is provided by comparing the analytical solution with the numerical one for a suite of numerical tests.

Sensitivity Analysis for Nonlinear Heat Conduction

2000 ◽  
Vol 123 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Kevin J. Dowding ◽  
Bennie F. Blackwell
Keyword(s):  
Heat Conduction ◽  
General Procedure ◽  
Model Parameters ◽  
Nonlinear Heat Conduction ◽  
Temperature Dependent ◽  
Sensitivity Equations ◽  
Verification Problem ◽  
Temperature Variable ◽  
Numerical Solver ◽  
Temperature Dependent Properties

Parameters in the heat conduction equation are frequently modeled as temperature dependent. Thermal conductivity, volumetric heat capacity, convection coefficients, emissivity, and volumetric source terms are parameters that may depend on temperature. Many applications, such as parameter estimation, optimal experimental design, optimization, and uncertainty analysis, require sensitivity to the parameters describing temperature-dependent properties. A general procedure to compute the sensitivity of the temperature field to model parameters for nonlinear heat conduction is studied. Parameters are modeled as arbitrary functions of temperature. Sensitivity equations are implemented in an unstructured grid, element-based numerical solver. The objectives of this study are to describe the methodology to derive sensitivity equations for the temperature-dependent parameters and present demonstration calculations. In addition to a verification problem, the design of an experiment to estimate temperature variable thermal properties is discussed.

scholarly journals The Robust Consensus of a Noisy Deffuant-Weisbuch Model

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jiangbo Zhang ◽  
Yiyi Zhao
Keyword(s):  
Discrete Time ◽  
Model Parameters ◽  
Positive Probability ◽  
Noise Strength ◽  
Opinion Formation ◽  
Time Step ◽  
Absolute Value ◽  
The Difference ◽  
Robust Consensus

We construct a new opinion formation of the Deffuant-Weisbuch model with the interference of the outer noise, where there are finite n agents and the evolution is discrete-time. The opinion interaction occurs by one randomly chosen pair at each time step. The difference to the original Deffuant-Weisbuch model is that communications of any selected pairs will be affected by noises. The aim of this paper is to study the robust consensus of this noisy Deffuant-Weisbuch model. We first define the noise strength as the maximum noise absolute value. We will then show that when the noise strength is less than a certain threshold, this noisy model will achieve T-robust consensus when t is sufficiently large; next we prove that the noisy model achieves robust consensus with a positive probability; finally, we demonstrate these results and provide numerical relations among the noise strength and some model parameters.

scholarly journals Thermal modeling of hydrogen storage by absorption in a magnesium hydrides tank

2014 ◽  
Vol 5 ◽  
pp. A21 ◽  
Author(s):  
Karim Lahmer ◽  
Rachid Bessaïh ◽  
Angel Scipioni ◽  
Mohammed El Ganaoui
Keyword(s):  
Hydrogen Absorption ◽  
Numerical Scheme ◽  
Computational Time ◽  
Governing Equations ◽  
Time Step ◽  
Kinetic Reaction ◽  
Fully Implicit ◽  
Temporal Profiles ◽  
Tank Geometry ◽  
Reaction Equations

This paper summarizes numerical results of hydrogen absorption simulated in an axisymmetric tank geometry containing magnesium hydride heated to 300 °C and at moderate storage pressure 1 MPa. The governing equations are solved with a fully implicit finite volume numerical scheme used by a commercial software FLUENT. The effect of the different kinetic reaction equations modeling hydrogen absorption was studied by the introduction of a specific subroutine at each time step in order to consider which one will provide results close to available experimental results. Spatial and temporal profiles of temperature and concentration in hydride bed are plotted. Results show that suitable method for our two-dimensional study is a CV-2D technique because it generates the smallest error especially during the beginning of the reaction. Also, its computational time is the shortest one compared to the other methods.

scholarly journals Robust Optimization Approach Using Scenario Concepts for Artillery Firing Scheduling Under Uncertainty

2019 ◽  
Vol 9 (14) ◽  
pp. 2811
Author(s):  
Choi ◽  
Yun ◽  
Kim ◽  
Jin ◽  
Kim
Keyword(s):  
Robust Optimization ◽  
Optimization Model ◽  
Numerical Experiments ◽  
Master Problem ◽  
Model Parameters ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Robust Solution ◽  
Reasonable Time ◽  
Scheduling Under Uncertainty

Real wars involve a considerable number of uncertainties when determining firing scheduling. This study proposes a robust optimization model that considers uncertainties in wars. In this model, parameters that are affected by enemy’s behavior and will, i.e., threats from enemy targets and threat time from enemy targets, are assumed as uncertain parameters. The robust optimization model considering these parameters is an intractable model with semi-infinite constraints. Thus, this study proposes an approach to obtain a solution by reformulating this model into a tractable problem; the approach involves developing a robust optimization model using the scenario concept and finding a solution in that model. Here, the combinations that express uncertain parameters are assumed by scenarios. This approach divides problems into master and subproblems to find a robust solution. A genetic algorithm is utilized in the master problem to overcome the complexity of global searches, thereby obtaining a solution within a reasonable time. In the subproblem, the worst scenarios for any solution are searched to find the robust solution even in cases where all scenarios have been expressed. Numerical experiments are conducted to compare robust and nominal solutions for various uncertainty levels to verify the superiority of the robust solution.

scholarly journals Tridimensional Long-Term Finite Element Analysis of Reinforced Concrete Structures with Rate-Type Creep Approach

2020 ◽  
Vol 10 (14) ◽  
pp. 4772
Author(s):  
Giovanni Di Luzio ◽  
Luigi Cedolin ◽  
Carlo Beltrami
Keyword(s):  
Finite Element ◽  
General Procedure ◽  
Constitutive Relations ◽  
Danube River ◽  
The Black Sea ◽  
Research Association ◽  
Time Step ◽  
Retardation Spectrum ◽  
Rate Type

This paper presents a general procedure for a rate-type creep analysis (based on the use of the continuous retardation spectrum) which avoids the need of recalculating the Kelvin chain stiffness elements at each time step. In this procedure are incorporated three different creep constitutive relations, two recommended by national codes such as the ACI (North-American) and EC2 (European) building codes and one by the RILEM research association. The approximate expressions of the different creep functions with the corresponding Dirichlet series are generated using the continuous retardation spectrum approach based on the Post–Widder formula. The proposed rate-type formulation is implemented into a 3D finite element code and applied to study the long-term deflections of a prestressed concrete bridge built in Romania, which crosses a wide artificial channel that connects the Danube river to the port of Constanta in the Black Sea.

Universal modification of vector weighted method of correlated sampling with finite computational cost

2019 ◽  
Vol 34 (1) ◽  
pp. 43-55
Author(s):  
Ilya N. Medvedev
Keyword(s):  
Markov Chain ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Radiation Transfer ◽  
Computational Cost ◽  
Linear Functionals ◽  
Weighted Estimator ◽  
Matrix Weights ◽  
Correlated Sampling

Abstract The weighted method of dependent trials or weighted method of correlated sampling (MCS) allows one to construct estimators for functionals based on the same Markov chain simultaneously for a given range of the problem parameters. Choosing an appropriate Markov chain, it is necessary to take into account additional conditions providing the finiteness of the computational cost of weighted MCS. In this paper we study the issue of finite computational cost of the method of correlated sampling (MCS) in application to evaluation of linear functionals of solutions to a set of systems of 2nd kind integral equations. A universal modification of the vector weighted MCS is constructed providing the branching of chain trajectory according to elements of matrix weights. It is proved that the computational cost of the constructed algorithm is bounded in the case the base functionals are also bounded. The results of numerical experiments using the modified weighted estimator are presented for some problems of the theory of radiation transfer subject to polarization.

scholarly journals Variational Inference over Nonstationary Data Streams for Exponential Family Models

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1942
Author(s):  
Andrés R. Masegosa ◽  
Darío Ramos-López ◽  
Antonio Salmerón ◽  
Helge Langseth ◽  
Thomas D. Nielsen
Keyword(s):  
Bayesian Inference ◽  
Variational Methods ◽  
Latent Variables ◽  
Exponential Family ◽  
Concept Drift ◽  
Variational Inference ◽  
Model Parameters ◽  
Time Step ◽  
Data Set ◽  
Family Models

In many modern data analysis problems, the available data is not static but, instead, comes in a streaming fashion. Performing Bayesian inference on a data stream is challenging for several reasons. First, it requires continuous model updating and the ability to handle a posterior distribution conditioned on an unbounded data set. Secondly, the underlying data distribution may drift from one time step to another, and the classic i.i.d. (independent and identically distributed), or data exchangeability assumption does not hold anymore. In this paper, we present an approximate Bayesian inference approach using variational methods that addresses these issues for conjugate exponential family models with latent variables. Our proposal makes use of a novel scheme based on hierarchical priors to explicitly model temporal changes of the model parameters. We show how this approach induces an exponential forgetting mechanism with adaptive forgetting rates. The method is able to capture the smoothness of the concept drift, ranging from no drift to abrupt drift. The proposed variational inference scheme maintains the computational efficiency of variational methods over conjugate models, which is critical in streaming settings. The approach is validated on four different domains (energy, finance, geolocation, and text) using four real-world data sets.

scholarly journals Effects of spatial discretization in ice-sheet modelling using the shallow-ice approximation

2006 ◽  
Vol 52 (176) ◽  
pp. 89-98 ◽  
Author(s):  
J. Van Den Berg ◽  
R.S.W. Van De Wal ◽  
J. Oerlemans
Keyword(s):  
Initial Conditions ◽  
Strong Dependence ◽  
Ice Sheet ◽  
Model Parameters ◽  
Time Step ◽  
Numerical Errors ◽  
Surface Gradient ◽  
Dimension Analysis ◽  
Sloping Bed ◽  
Ice Model

AbstractThis paper assesses a two-dimensional, vertically integrated ice model for its numerical properties in the calculation of ice-sheet evolution on a sloping bed using the shallow-ice approximation. We discuss the influence of initial conditions and individual model parameters on the model’s numerical behaviour, with emphasis on varying spatial discretizations. The modelling results suffer badly from numerical problems. They show a strong dependence on gridcell size and we conclude that the widely used gridcell spacing of 20 km is too coarse. The numerical errors are small in each single time-step, but increase non-linearly over time and with volume change, as a result of feedback of the mass balance with height. We propose a new method for the calculation of the surface gradient near the margin, which improves the results significantly. Furthermore, we show that we may use dimension analysis as a tool to explain in which situations numerical problems are to be expected.

scholarly journals A Spring–Dashpot System for Modelling Lung Tumour Motion in Radiotherapy

2010 ◽  
Vol 11 (1) ◽  
pp. 13-26 ◽  
Author(s):  
P. L. Wilson ◽  
J. Meyer
Keyword(s):  
Numerical Scheme ◽  
3D Model ◽  
Tracking System ◽  
Spatial Relationship ◽  
Single Equation ◽  
Lung Tumour ◽  
Model Parameters ◽  
Optimization Routine ◽  
Parameter Ranges ◽  
Differential Equations Models

A 3D system of springs and dashpots is presented to model the motion of a lung tumour during respiration. The main guiding factor in configuring the system is the spatial relationship between abdominal and lung tumour motion. A coupled, non-dimensional triple of ordinary differential equations models the tumour motion when driven by a 3D breathing signal. Asymptotic analysis is used to reduce the system to a single equation driven by a 3D signal, in the limit of small lateral and transverse tumour motions. A numerical scheme is introduced to solve this equation, and tested over wide parameter ranges. Real clinical data is used as input to the model, and the tumour motion output is in excellent agreement with that obtained by a prototype tumour tracking system, with model parameters obtained by optimization. The fully 3D model has the potential to accurately model the motion of any lung tumour given an abdominal signal as input, with model parameters obtained from an internal optimization routine.

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