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A Numerical Study for a Flexible Euler-Bernoulli Beam with a Force Control in Velocity and a Moment Control in Rotating Velocity
In this paper, we numerically study a flexible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity. First, we show the existence and uniqueness of the weak solution using Faedo-Galerkin’s method with the intermediate spaces. Then, we use the finite elements method with the cubic Hermite polynomials for the approximation of (1.1)–(1.5) in space such that the semi-discrete scheme obtained is stable and convergent. In addition, an a-priori error estimate is obtained. Finally, we perform numerical simulations in order to validate this method.
Beam Equation, Existence and Uniqueness, Higher Regularity, Finite Element Method, Galerkin Method, Priori Estimates.
- Paul J. Allen, A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Soc., 21 (1969), 412–416.
- Shahabaddin Ebrahimi Atani, The ideal theory in quotients of commutative semirings, Glasnik Matematicki, 42 (62)(2007), 301 - 308.
- A. P. Goh Abro, J. M. Gossrin Bomisso, A. Kidjgbo Tour, and Adama Coulibaly, A Numerical Method by Finite Element Method (FEM) of an Euler-Bernoulli beam to Variable Coefficients, Advances in Mathematics: Scientific Journal, 9 (2020), 8485 - 8510.
- H. T. Banks and I. G. Rosen, Computational methods for the identification of spatially varying stiffness and damping in beams, Theory and advanced technology, 3 (1987), 1 - 32.
- M. Basson and N. F. J. Van Rensburg, Galerkin finite element approximation of general linear second order hyperbolic equations, Numer. Func. Anal. Opt. 34(9) (2013), 976 - 1000.
- M. Basson, B. Stapelberg and N. F. J. Van Rensburg, Error Estimates for Semi-Discrete and Fully Discrete Galerkin Finite Element Approximations of the General Linear Second-Order Hyperbolic Equation, Numerical Functional Analysis and Optimization, 38(4) (2017), 466 - 485.
- Bomisso G. Jean Marc, Tour´e K. Augustin and Yoro Gozo, Dissipative Numerical Method for a Flexible Euler-Bernoulli Beam with a Force Control in Rotation and Velocity Rotation, Journal of Mathematics Research, 9 (2017), 30 - 48.
- H. Brezis, Fonctional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
- L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
- J. L. Lions and E. Magenes, Probl`emes aux limites non homog`enes et Applications, Dunod, 1 1968.
- Z. H. Luo, B. Z. Guo and O. Morgul, Stability and stabilization of infinite dimensional systems with applications, Communications and control Engineering series, Springer-Verlag London Ltd, London, 1999.
- Mensah E. Patrice, On the numerical approximation of the spectrum of a flexible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity, Far East Journal of Mathematical Sciences (FJMS), 126(1)(2020), 13 - 31.
- M. Miletic and A. Arnold, A piezoelectric Euler-Bernoulli beam with dynamic boundary control: Stability and dissipative FEM, Acta Applicandae Mathematicae, (2014), 1 - 37.
- B. Rao, Uniform stabilization of a hybrid system of elasticity, Siam J. Control and Optimization 33(2) (1995),440 - 454.
- A. Shkalikov, Boundary problem for ordinary differential operators with parameter in boundary conditions, Journal of Soviet Mathematics, 33 (1986), 1311 - 1342.
- R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, Springer-Verlag, New York, 68 1988.
- Tzin Wang, A Hermite cubic immersed finite element space for beam design problems, Thesis, Blacksburg Virginia, 2005.
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