Abstract
In this paper, for the initial and boundary value problem of beams with structural damping, by introducing intermediate variables, the original fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed. Numerical examples are provided to confirm the theoretical results. In the end, we test the value of δ to observe its influence on the model.
Highlights
The beam is the most important part of the upper frame of a building, which is widely used in engineering projects, bridge construction and aerospace
For the initial and boundary value problem of beams with structural damping, by introducing intermediate variables, the original fourthorder problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed
The spatial derivative term is discretized by the mixed finite volume element method, and the time derivative term is discretized by the backward Euler scheme to construct the mixed finite volume element scheme of (3); in Section 3, some necessary lemmas are given; in Section 4, we prove the existence and uniqueness of solutions of mixed finite volume element scheme; The convergence of semi-discrete and fully-discrete mixed finite volume element schemes is proved in Sections 5 and 6 respectively; in Section 7, some numerical examples are given to verify the accuracy of the scheme, which indicates that the scheme has high practicability
Summary
The beam is the most important part of the upper frame of a building, which is widely used in engineering projects, bridge construction and aerospace. Fan studied the existence, uniqueness and regularity of the abstract model for the vibration equations of the beam with structural damping and the existence of the global mild solution in literature [17] [18]. The objective of this paper is to construct a mixed finite volume element scheme for the vibration problem of the beam with structural damping (3). This scheme was first proposed by Russell [20] in 1995 when he solved the second-order elliptic problem. Wang [23] studied the equilibrium equation of beams by using the mixed finite volume element method and proved that the scheme has first-order accuracy in the discrete H1 half-norm and the discrete L2 norm. The spatial derivative term is discretized by the mixed finite volume element method, and the time derivative term is discretized by the backward Euler scheme to construct the mixed finite volume element scheme of (3); in Section 3, some necessary lemmas are given; in Section 4, we prove the existence and uniqueness of solutions of mixed finite volume element scheme; The convergence of semi-discrete and fully-discrete mixed finite volume element schemes is proved in Sections 5 and 6 respectively; in Section 7, some numerical examples are given to verify the accuracy of the scheme, which indicates that the scheme has high practicability
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