scholarly journals Algorithmic differentiation improves the computational efficiency of OpenSim-based optimal control simulations of movement

2019 ◽  
Author(s):  
Antoine Falisse ◽  
Gil Serrancolí ◽  
Christopher L. Dembia ◽  
Joris Gillis ◽  
Friedl De Groote
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Three Dimensional ◽  
Second Order ◽  
Computational Time ◽  
Code Transformation ◽  
Algorithmic Differentiation ◽  
Implicit Differential Equations ◽  
Direct Collocation ◽  
Source Code Transformation

AbstractAlgorithmic differentiation (AD) is an alternative to finite differences (FD) for evaluating function derivatives. The primarily aim of this study was to demonstrate the computational benefits of using AD instead of FD in OpenSim-based optimal control simulations. The secondary aim was to evaluate computational choices including different AD tools, different linear solvers, and the use of first- or second-order derivatives. First, we enabled the use of AD in OpenSim through a custom source code transformation tool and through the operator overloading tool ADOL-C. Second, we developed an interface between OpenSim and CasADi to perform optimal control simulations. Third, we evaluated computational choices through simulations of perturbed balance, two-dimensional predictive simulations of walking, and three-dimensional tracking simulations of walking. We performed all simulations using direct collocation and implicit differential equations. Using AD through our custom tool was between 1.8 ± 0.1 and 17.8 ± 4.9 times faster than using FD, and between 3.6 ± 0.3 and 12.3 ± 1.3 times faster than using AD through ADOL-C. The linear solver efficiency was problem-dependent and no solver was consistently more efficient. Using second-order derivatives was more efficient for balance simulations but less efficient for walking simulations. The walking simulations were physiologically realistic. These results highlight how the use of AD drastically decreases computational time of optimal control simulations as compared to more common FD. Overall, combining AD with direct collocation and implicit differential equations decreases the computational burden of optimal control simulations, which will facilitate their use for biomechanical applications.

THREE-DIMENSIONAL STAGNATION-POINT FLOW OVER A PERMEABLE STRETCHING/SHRINKING SHEET WITH A SECOND ORDER SLIP FLOW

2021 ◽  
Vol 10 (9) ◽  
pp. 3273-3282
Author(s):  
M.E.H. Hafidzuddin ◽  
R. Nazar ◽  
N.M. Arifin ◽  
I. Pop
Keyword(s):  
Differential Equations ◽  
Stagnation Point ◽  
Flow Model ◽  
Nonlinear Partial Differential Equations ◽  
Slip Flow ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Second Order ◽  
Stagnation Point Flow ◽  
Shrinking Sheet

The problem of steady laminar three-dimensional stagnation-point flow on a permeable stretching/shrinking sheet with second order slip flow model is studied numerically. Similarity transformation has been used to reduce the governing system of nonlinear partial differential equations into the system of ordinary (similarity) differential equations. The transformed equations are then solved numerically using the \texttt{bvp4c} function in MATLAB. Multiple solutions are found for a certain range of the governing parameters. The effects of the governing parameters on the skin friction coefficients and the velocity profiles are presented and discussed. It is found that the second order slip flow model is necessary to predict the flow characteristics accurately.

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

A Numerical Model to Predict the Thermal and Psychrometric Response of Electronic Packages

1999 ◽  
Vol 123 (3) ◽  
pp. 200-210 ◽  
Author(s):  
J. V. C. Vargas ◽  
G. Stanescu ◽  
R. Florea ◽  
M. C. Campos
Keyword(s):  
Differential Equations ◽  
Physical Model ◽  
Finite Volume ◽  
Three Dimensional ◽  
Computational Time ◽  
Finite Volume Scheme ◽  
Electronic Packages ◽  
Simulation Design ◽  
Experimental Conditions ◽  
Volume Scheme

This paper introduces a general computational model for electronic packages, e.g., cabinets that contain electronic equipment. A simplified physical model, which combines principles of classical thermodynamics and heat transfer, is developed and the resulting three-dimensional differential equations are discretized in space using a three-dimensional cell centered finite volume scheme. Therefore, the combination of the proposed simplified physical model with the adopted finite volume scheme for the numerical discretization of the differential equations is called a volume element model (VEM). A typical cabinet was built in the laboratory, and two different experimental conditions were tested, measuring the temperatures at forty-six internal points. The proposed model was utilized to simulate numerically the behavior of the cabinet operating under the same experimental conditions. Mesh refinements were conducted to ensure the convergence of the numerical results. The converged mesh was relatively coarse (504 cells), therefore the solutions were obtained with low computational time. The model temperature results were directly compared to the steady-state experimental measurements of the forty-six internal points, with good quantitative and qualitative agreement. Since accuracy and low computational time are combined, the model is shown to be efficient and could be used as a tool for simulation, design, and optimization of electronic packages.

scholarly journals Field lines of gravity, their curvature and torsion, the Lagrange and the Hamilton equations of the plumbline

1997 ◽  
Vol 40 (5) ◽  
Author(s):  
E. W. Grafarend
Keyword(s):  
Differential Equations ◽  
Three Dimensional ◽  
Physical Geodesy ◽  
Equipotential Surface ◽  
Second Order ◽  
Gravity Potential ◽  
Problem Solver ◽  
Generalized Force ◽  
Field Lines ◽  
Curvature And Torsion

The length of the gravitational field lines/of the orthogonal trajectories of a family of gravity equipotential surfaces/of the plumbline between a terrestrial topographic point and a point on a reference equipotential surface like the geoid í also known as the orthometric height í plays a central role in Satellite Geodesy as well as in Physical Geodesy. As soon as we determine the geometry of the Earth pointwise by means of a satellite GPS (Global Positioning System: «global problem solver») we are left with the problem of converting ellipsoidal heights (geometric heights) into orthometric heights (physical heights). For the computation of the plumbline we derive its three differential equations of first order as well as the three geodesic equations of second order. The three differential equations of second order take the form of a Newton differential equation when we introduce the parameter time via the Marussi gauge on a conformally flat three-dimensional Riemann manifold and the generalized force field, the gradient of the superpotential, namely the modulus of gravity squared and taken half. In particular, we compute curvature and torsion of the plumbline and prove their functional relationship to the second and third derivatives of the gravity potential. For a spherically symmetric gravity field, curvature and torsion of the plumbline are zero, the plumbline is straight. Finally we derive the three Lagrangean as well as the six Hamiltonian differential equations of the plumbline, in particular in their star form with respect to Marussi gauge.

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