Richard courant and the finite element method: A further look

Williamson, Frank R.

doi:10.1016/0315-0860(80)90001-4

Cited by 13 publications

(6 citation statements)

References 9 publications

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“…Solving the system of equations ( 10) minimizes the performance index (6). Note that q i and u i are independent variables and that 4n boundary conditions are required.…”

Section: B Optimal Controlmentioning

confidence: 99%

“…Again, this behavior was observed for both joints in our simulations. This occurs because our optimization method focuses in minimizing a performance index (6) for which the integrand is a function of the joint torques. It is therefore natural that torques oscillate around the zero value.…”

Section: Optimal Control Of a 2-dof Robotic Manipulatormentioning

confidence: 99%

“…The finite element method (FEM) has become a favored technique for solving engineering problems relating to the estimation of stress, strain or wear in elastic materials (see references [1]- [5] for some examples). Its origins can be linked to the works of Richard Courant back in 1943 [6]. However, it has been advocated that the FEM can easily be used as a general approximation method for solving boundary-value problems that arise in science and engineering applications [7], [8].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Control of Robotic Systems Using Finite Elements for Time Integration of Covariant Control Equations

2021

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We used an optimal control method involving covariant control equations as optimality conditions, to command the actuators of robot manipulators. These form a coupled system of second order nonlinear ordinary differential equations when associated with the robot motion equations. By solving this system, the control action required to take the robot from an initial to a final state is optimized in a prescribed time. However, the target set of equations exhibited stiffness. Therefore, an adequate solution could only be found for short trajectory durations with readily available numerical methods. We examined a time discretization procedure based on cubic and quintic Hermite finite elements which exhibited superconvergence properties for interpolation. This motivated us to develop a time integration algorithm based on Hermite's technique, where motion and control equations were perturbed to solve the optimal control problem. The optimal motion of a robotic manipulator was simulated using this algorithm. Our method was compared with a commercial differential equations solver on the basis of specific indicators. It outperformed the commercial solver by effectively solving the stiff set of equations for longer trajectory durations, with the cubic elements performing better than the quintic ones in this sense. The convergence analysis of our method confirmed that the quintic elements are more precise at the cost of increased computational burden, but converge at a lower rate than expected. Controlled motion experiments on a robotic manipulator validated our methodology. Trajectories were smoothly tracked and results exposed further methodology improvements.

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“…Solving the system of equations ( 10) minimizes the performance index (6). Note that q i and u i are independent variables and that 4n boundary conditions are required.…”

Section: B Optimal Controlmentioning

confidence: 99%

Section: Optimal Control Of a 2-dof Robotic Manipulatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal Control of Robotic Systems Using Finite Elements for Time Integration of Covariant Control Equations

2021

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“…Development of the finite element method in structural mechanics is usually based on an energy principle such as the virtual work principle or the minimum total potential energy principle. The finite element analysis from the mathematical side was first developed in 1943 by Richard Courant, who used the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems [1]. From the engineering side, the finite element analysis originated as the displacement method of the matrix structural analysis, which emerged over the course of several decades mainly in British aerospace research as a variant suitable for computers.…”

Section: Introductionmentioning

confidence: 99%

Model for Deflection Analysis in Cantilever Beam

Oladejo¹,

Abu²,

Bamiro³

2018

EJERS

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Exposure to Finite Element Method is beneficial to undergraduate engineering students; and educators have an obligation to introduce students to modern engineering tools. However, teaching of the course is computational intensive and existing propriety software are very expensive. Different approaches to introducing students to the FEM have been proposed. Existing approaches make use of learning modules of commercial Finite Element Analysis (FEA) packages such as ANSYS, ABAQUS, COMSOL Multiphysics, ALGOR, JL AutoFEA Analyzer and PRO/MECHANICA. This paper presents an in-house developed finite-element-based computer model via the virtual work principle using Linear Strain Triangular (LST) elements for deflection analysis on cantilever beam. The validation and capability characteristics was demonstrated by applying it to a cantilever beam, subjected to a point load, using both coarse (4-element) and fine (10-element) meshes. The model gave results very close to those obtained analytically; the 10-element mesh gave better results than the 4-element mesh. The model has made the analysis more flexible, and also made visualization and presentation of results easier for better judgment. It facilitates the presentation of the basic rules which govern FEA and helps in the learning of different aspects of the numerical technique; hence the model will serve as reliable tool in undergraduate engineering programme

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“…The use of finite element analysis is not limited to the engineering field, as there are also medical and geospatial applications. The early development of the finite element can be traced back to the work of Courant [1], followed by the work of Martin [2], which applied the solutions for structural analyses at Boeing Company in the 1950s. Further work by Argyris, Clough, Turner, and Zienkiewicz developed the governing mathematical equation for the finite element method [3].…”

Section: Introductionmentioning

confidence: 99%