scholarly journals Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed-form of Richards' equation

2016 ◽  
Author(s):  
Fadji Hassane Maina ◽  
Philippe Ackerer
Keyword(s):  
Adaptive Algorithm ◽  
Water Saturation ◽  
The State ◽  
Richards Equation ◽  
Time Step ◽  
Mixed Form ◽  
Convergence Error ◽  
Time Stepping ◽  
State Variable ◽  
Newton Raphson

Abstract. The solution of the mathematical model for flow in variably saturated porous media described by Richards equation (RE) is subject to heavy numerical difficulties due to its highly non-linear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed-form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the time step magnitude within the iteration procedure while the state variable is kept constant. The Ross method is an example of this type of scheme, and we show that it is equivalent to the Newton-Raphson method with a time-adaptive algorithm. Both algorithms are coupled to different time stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change of water saturation. Three different test cases are used to evaluate the efficiency of these algorithms. The numerical results highlight the necessity of implementing two types of errors: the iterative convergence error (maximum difference of the state variable between two iterations) and an estimate of the time truncation errors. The algorithms using these two types of errors together were found to be the most efficient when highly accurate results are required.

scholarly journals Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

2017 ◽  
Vol 21 (6) ◽  
pp. 2667-2683 ◽  
Author(s):  
Fadji Hassane Maina ◽  
Philippe Ackerer
Keyword(s):  
Water Saturation ◽  
Richards Equation ◽  
Time Step ◽  
Mixed Form ◽  
Nonlinear Properties ◽  
Truncation Errors ◽  
Time Stepping ◽  
Highly Nonlinear ◽  
Newton Raphson ◽  
The Mathematical Model

Abstract. The solution of the mathematical model for flow in variably saturated porous media described by the Richards equation (RE) is subject to heavy numerical difficulties due to its highly nonlinear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the time-step magnitude within the iteration procedure while the nonlinear parameters are computed with the state variable at the previous time. The Ross method is an example of this type of scheme, and we show that it is equivalent to the Newton–Raphson method with a time-adaptive algorithm.Both algorithms are coupled to different time-stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change in water saturation. Three different test cases are used to evaluate the efficiency of these algorithms.The numerical results highlight the necessity of implementing an estimate of the time truncation errors.

Application of the Extended Kalman Filter to Control of a Shape Memory Alloy Arm

2003 ◽  
Author(s):  
Mohammad H. Elahinia ◽  
Hashem Ashrafiuon ◽  
Mehdi Ahmadian ◽  
Daniel J. Inman
Keyword(s):  
Kalman Filter ◽  
Shape Memory Alloy ◽  
Shape Memory ◽  
Extended Kalman Filter ◽  
State Vector ◽  
Angular Position ◽  
The State ◽  
State Variables ◽  
Time Step ◽  
State Variable

This paper presents a robust nonlinear control that uses a state variable estimator for control of a single degree of freedom rotary manipulator actuated by Shape Memory Alloy (SMA) wire. A model for SMA actuated manipulator is presented. The model includes nonlinear dynamics of the manipulator, a constitutive model of the Shape Memory Alloy, and the electrical and heat transfer behavior of SMA wire. The current experimental setup allows for the measurement of only one state variable which is the angular position of the arm. Due to measurement difficulties, the other three state variables, arm angular velocity and SMA wire stress and temperature, cannot be directly measured. A model-based state estimator that works with noisy measurements is presented based on the Extended Kalman Filter (EKF). This estimator predicts the state vector at each time step and corrects its prediction based on the angular position measurements. The estimator is then used in a nonlinear and robust control algorithm based on Variable Structure Control (VSC). The VSC algorithm is a control gain switching technique based on the arm angular position (and velocity) feedback and EKF estimated SMA wire stress and temperature. The state vector estimates help reduce or avoid the undesirable and inefficient overshoot problem in SMA one-way actuation control.

scholarly journals A Temperature-based Controller for a Shape Memory Alloy Actuator

2005 ◽  
Vol 127 (3) ◽  
pp. 285-291 ◽  
Author(s):  
Mohammad H. Elahinia ◽  
Hashem Ashrafiuon ◽  
Mehdi Ahmadian ◽  
Hanghao Tan
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory ◽  
State Vector ◽  
Angular Position ◽  
Variable Structure ◽  
The State ◽  
State Variables ◽  
Time Step ◽  
State Variable ◽  
Control Gain

This paper presents a robust nonlinear control that uses a state variable estimator for control of a single degree of freedom rotary manipulator actuated by shape memory alloy (SMA) wire. A model for SMA actuated manipulator is presented. The model includes nonlinear dynamics of the manipulator, a constitutive model of the shape memory alloy, and the electrical and heat transfer behavior of SMA wire. The current experimental setup allows for the measurement of only one state variable which is the angular position of the arm. Due to measurement difficulties, the other three state variables, arm angular velocity and SMA wire stress and temperature, cannot be directly measured. A model-based state estimator that works with noisy measurements is presented based on the extended Kalman filter (EKF). This estimator estimates the state vector at each time step and corrects its estimation based on the angular position measurements. The estimator is then used in a nonlinear and robust control algorithm based on variable structure control (VSC). The VSC algorithm is a control gain switching technique based on the arm angular position (and velocity) feedback and EKF estimated SMA wire stress and temperature. Using simulation it is shown that the state vector estimates help reduce or avoid the undesirable and inefficient overshoot problem in SMA one-way actuation control.

scholarly journals Implicit and semi-implicit second-order time stepping methods for the Richards equation

2021 ◽  
Vol 148 ◽  
pp. 103841
Author(s):  
Sana Keita ◽  
Abdelaziz Beljadid ◽  
Yves Bourgault
Keyword(s):  
Second Order ◽  
Richards Equation ◽  
Time Stepping

A Self-adaptive Algorithm with Newton Raphson Method for Parameters Identification of Photovoltaic Modules and Array

2021 ◽  
Author(s):  
Patrick Juvet Gnetchejo ◽  
Salomé Ndjakomo Essiane ◽  
Abdouramani Dadjé ◽  
Pierre Ele ◽  
Daniel Eutyche Mbadjoun Wapet ◽  
...  
Keyword(s):  
Adaptive Algorithm ◽  
Parameters Identification ◽  
Photovoltaic Modules ◽  
Newton Raphson ◽  
Raphson Method ◽  
Self Adaptive

scholarly journals Exploring the Limits of Floating-Point Resolution for Hardware-In-the-Loop Implemented with FPGAs

Electronics ◽  
2018 ◽  
Vol 7 (10) ◽  
pp. 219 ◽  
Author(s):  
Alberto Sanchez ◽  
Elías Todorovich ◽  
Angel de Castro
Keyword(s):  
The State ◽  
Floating Point ◽  
Hardware In The Loop ◽  
State Variables ◽  
Numerical Resolution ◽  
Digital Devices ◽  
State Variable ◽  
Simulation Step ◽  
Field Programmable ◽  
The Difference

As the performance of digital devices is improving, Hardware-In-the-Loop (HIL) techniques are being increasingly used. HIL systems are frequently implemented using FPGAs (Field Programmable Gate Array) as they allow faster calculations and therefore smaller simulation steps. As the simulation step is reduced, the incremental values for the state variables are reduced proportionally, increasing the difference between the current value of the state variable and its increments. This difference can lead to numerical resolution issues when both magnitudes cannot be stored simultaneously in the state variable. FPGA-based HIL systems generally use 32-bit floating-point due to hardware and timing restrictions but they may suffer from these resolution problems. This paper explores the limits of 32-bit floating-point arithmetics in the context of hardware-in-the-loop systems, and how a larger format can be used to avoid resolution problems. The consequences in terms of hardware resources and running frequency are also explored. Although the conclusions reached in this work can be applied to any digital device, they can be directly used in the field of FPGAs, where the designer can easily use custom floating-point arithmetics.

Fast Conjugate Heat Transfer Simulation of Long Transient Flexible Operations Using Adaptive Time Stepping

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
R. Maffulli ◽  
L. He ◽  
P. Stein ◽  
G. Marinescu
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Time Derivative ◽  
Computational Cost ◽  
Strong Impact ◽  
Time Step ◽  
Time Stepping ◽  
Adaptive Time ◽  
Fully Coupled ◽  
Adaptive Time Stepping

The emerging renewable energy market calls for more advanced prediction tools for turbine transient operations in fast startup/shutdown cycles. Reliable numerical analysis of such transient cycles is complicated by the disparity in time scales of the thermal responses in fluid and solid domains. Obtaining fully coupled time-accurate unsteady conjugate heat transfer (CHT) results under these conditions would require to march in both domains using the time-step dictated by the fluid domain: typically, several orders of magnitude smaller than the one required by the solid. This requirement has strong impact on the computational cost of the simulation as well as being potentially detrimental to the accuracy of the solution due to accumulation of round-off errors in the solid. A novel loosely coupled CHT methodology has been recently proposed, and successfully applied to both natural and forced convection cases that remove these requirements through a source-term based modeling (STM) approach of the physical time derivative terms in the relevant equations. The method has been shown to be numerically stable for very large time steps with adequate accuracy. The present effort is aimed at further exploiting the potential of the methodology through a new adaptive time stepping approach. The proposed method allows for automatic time-step adjustment based on estimating the magnitude of the truncation error of the time discretization. The developed automatic time stepping strategy is applied to natural convection cases under long (2000 s) transients: relevant to the prediction of turbine thermal loads during fast startups/shutdowns. The results of the method are compared with fully coupled unsteady simulations showing comparable accuracy with a significant reduction of the computational costs.

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