Abstract: Fundamentals based on periodic structure in Solid-State Physics are applied to seismic isolation and propagation in periodic structure.
Firstly, a compound periodic steel structure parameterized by its phase differences between cells is constructed: SupposeIt is supposed that a steel structure contains no less than 2 orientable periodic cells, and these cells are glued together one by one. Beginning from the fixed first cell, by isometric mapping composed of gluing and orientation maps, it aligned with the 2nd cell forms a 2-cell periodic structure with positive phase difference between their orientations, then the 2-cell structure extends at its both ends into an 1-dimensional periodic structure containing at least 2 orientable periodic substructures. In this 1-dimensional compound periodic structure, each cell further extends on plane, or into space, according to isometric mapping composed of gluing and orientation maps, and shapes a 2-dimensioanal, and those compound periodic structures of dimensions bigger than 2. Composition of gluing maps with orientations plays the mapping role at those common parts of periodic structures glued together one by one, propagates seismic wave from the beginning cell to the next cell, in turn seismic wave ranges through the integrated compound periodic structure.
Secondly, seismic response parameterized by wave number, number of gluing mappings, scale of the structure, and in these parametersphase differences in the structures is computed in compound periodic structure. By linear response assumption, response with each of parameters is computed respectively at first, then total response of compound periodic structure is obtain from their summation.
Seismic wave is stochastic, and obeys a given stochastic differential equation, and then structural response is also set of values with probability. Seismic responses in 2-cell and 1-dimensional periodic structure are discussed at first. In 2-cell compound periodic steel structure, the given stochastic differential equation is input at its first cell, under certain starting condition and boundary condition, and with the help of set-valued stochastic process and martingale knowledge, seismic response in the first cell is calculated; further by phase difference and trigonometric function operations, seismic responses in the second cell of 2-cell structure and each cell in 1-dimensional structure are calculated.
Finally, Markov controlling to structural seismic response is performed to select the optimal phase angle, and realize a sort of polynomial response goal: according to the set-valued stochastic process property of the response in periodic cell, it is represented by a diffusion process, and its infinitesimal generator is used to construct the corresponding Dirichlet-Poisson equation, further the corresponding Hamilton-Jacobi-Bellman（HJB）equation is constructed to solve the optimal phase angle, substituted back into the Dirichlet-Poisson equation to solve the controlling result. The result decreases as propagating time and initial amplitude of seismic wave increase. Simulating the given stochastic seismic wave by Monte Carlo method, and generating a stationary distribution, then sampling the given stochastic seismic wave by this distribution, calculation of variances of the sampling series and the Markov controlling result and comparison of expectations of indicator function of the given stochastic seismic wave and the Markov controlling result show that variance of the Markov controlling result is smaller than that of sampling series, what is more, the Markov controlling deceases expectation of the given stochastic seismic wave.
Realization of both representation to seismic response in compound periodic steel structure by stochastic equation and its Markov controlling is in servitude to reference for its stochastic prediction.