Polynomial eigenvalue problems are typically solved by linearization, solving the linear problem with a black-box code. However, when the leading or trailing coefficients are singular, they may contribute infinite or zero eigenvalues. The fact that there are infinite/zero eigenvalues may not be detected by a black-box code, which may just return very large or small numbers in finite-precision arithmetic. In this talk we will consider initially deflating these infinite and/or zero eigenvalues to leave a smaller problem that can be handled by a black-box code. We will look at which linearizations are suitable and how the deflated problem can be put in a suitable form to be handled by the QZ algorithm as implemented in LAPACK.
I intend to give a pratice run of a talk I have to give to non-mathematicians on `Uncertainty Quantification.' I will talk in particular, about the challenge of solving linear systems in engineering problems with random inputs. There will be lots of pretty pictures but no PDEs, probability spaces, or scary maths words.
Noise reduction treatments in the form of compliant coatings are commonly applied to vibrating underwater structures in order to reduce the level of sound they radiate. Such coatings can be highly effective but if not continuous the exposed underlying vibrating structure may raise radiated noise subatantially. Some results from an ongoing analysis of a 2D problem with a small rectangular defect are presented.
We study propagation of strain waves in non-linear hyperelastic
media with microstructure. As an illustrative example a 1D model of a
layered composite material is considered. Geometrical non-linearity is
predicted adopting refined relations between the strains and the gradients
of displacements. Physical non-linearity is described using the Murnaghan
elastic potential. The constitutive wave equation is derived by the
higher-order asymptotic homogenization method. Magnitudes of the
coefficients at the dispersive terms are determined by the Floquet-Bloch
approach. In the case of weak non-linearity, an asymptotic solution is
developed. In the case of strong non-linearity, the method of direct
integration in elliptic functions is applied. As the results, explicit
analytical expressions for stationary quasi-linear, cnoidal and solitary
waves are obtained. A number of non-linear phenomena are detected, such as
generation of higher-order modes, multi-mode energy pumping and
localization. Numerical results are given and practical relevancy of the
above mentioned effects is discussed.