![]() ![]() Daraio, “Optimal design of composite granular protectors,” Mechanics of Advanced Materials and Structures, vol. ![]() Sen, “Decorated, tapered, and highly nonlinear granular chain,” Phys. Jin, “Energy trapping and shock disintegration in a composite granular medium,” Phys. Willenbring, Trilinos users guide, United States. Kuznetsov, “MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs,” ACM Transactions on Mathematical Software (TOMS), vol. Doedel, “AUTO: A program for the automatic bifurcation analysis of autonomous systems,” Congr. Doney, “Solitary waves in the granular chain,” Physics Reports, vol. Nesterenko, Dynamics of heterogeneous materials, Springer, 2001. The EF/DDDAS protocols presented here are, therefore, a step toward general-purpose protocols for performing detailed bifurcation analyses directly on laboratory experiments, not only on their mathematical models, but also on measured data. ![]() In particular, the focus is on the detection/stability analysis of time-periodic, spatially localized structures referred to as “dark breathers.” Results in this chapter highlight, both experimentally and numerically, that the number of breathers can be controlled by varying the frequency as well as the amplitude of an “out-of-phase” actuation, and that a “snaking” structure in the bifurcation diagram (computed through standard, model-based numerical methods for dynamical systems) is also recovered through the EF/DDDAS methods operating on a black-box simulator. An illustrative example demonstrates the experimental realization of a chain of granular particles (a so-called engineered granular chain). Necessary.This chapter discusses the development and implementation of algorithms based on Equation-Free/Dynamic Data-Driven Applications Systems (EF/DDDAS) protocols for the computer-assisted study of the bifurcation structure of complex dynamical systems, such as those that arise in biology (neuronal networks, cell populations), multiscale systems in physics, chemistry, and engineering, and system modeling in the social sciences. How to provide additional parameters to the function mfun, if To use a function handle, first create a function with the signatureįunction y = mfun(x,opt). Preconditioner matrix, making the calculation more efficient. Handle performs matrix-vector operations instead of forming the entire M2 as function handles instead of matrices. You can optionally specify any of M, M1, or Lsqr treats unspecified preconditioners as identity For more information on preconditioners, see Iterative Methods for Linear Systems. You also can use equilibrate prior to factorization to improve the condition number of Ilu and ichol to generate preconditioner matrices. Square coefficient matrices, you can use the incomplete matrix factorization functions System and make it easier for lsqr to converge quickly. You can specify a preconditioner matrix M or its matrixįactors M = M1*M2 to improve the numerical aspects of the linear Preconditioner matrices, specified as separate arguments of matrices or function In MATLAB®, write a function that creates these vectors and adds them together, thus giving the value of A*x or A'*x, depending on the flag input: Likewise, the expression for A T x becomes:Ī T x =. The resulting vector can be written as the sum of three vectors:Ī x = + + 2 ⋅. The nonzero elements in the result correspond with the nonzero tridiagonal elements of A.Ī x =. When A multiplies a vector, most of the elements in the resulting vector are zeros. Since this tridiagonal matrix has a special structure, you can represent the operation A*x with a function handle.
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