Fractional Order Models for Vibration and Wave Propagation Applications

John Hollkamp, Purdue University

Abstract

The continuously growing interest in the design and synthesis of heterogeneous structures has further highlighted the need for accurate modeling and efficient simulation techniques. In the context of structural dynamics and elastic wave propagation simulations, theoretical models can be broadly divided into two main categories: discrete and continuous. Discrete parameter models, such as finite difference or finite element analysis, not only provide a simplified representation of complex systems, but also are some of the most powerful structural computational techniques available to-date. In the case of geometrically complex and heterogeneous structures, this class of techniques typically produces numerical models involving a large number of degrees of freedom, ultimately leading to significant computational times and resources. Historically, model order reduction techniques have been one of the most powerful tools to reduce the number of degrees of freedom while maintaining high levels of accuracy and fidelity. On the other hand, continuous parameter models can provide a more accurate and concise representation of the actual physical system, but the underlying mathematical formulation (typically based on partial differential equations with variable coefficients) is typically not well suited to analytical solutions, especially for systems containing complex geometries and boundary conditions. Homogenization techniques are an important class of models that can overcome some of these complexities while still preserving the ability to provide a concise mathematical representation and, possibly, analytical closed-form solutions. Despite the significant advancements and the many successes that the engineering community has achieved in the development of these two classes of methods, both categories still encounter various limitations including, but not limited to, narrow-band frequency accuracy, applicability in the long wavelength regime, and still potentially expensive numerical evaluations. The emergent mathematical field of fractional calculus - the calculus of integrals and derivatives of any real or complex order - provides an excellent opportunity to develop novel, accurate, and efficient models for simulations of heterogeneous structures. Fractional operators, possessing characteristics such as memory effects, nonlocality, multi-scale capabilities, and hybrid behavior, can provide advanced mathematical tools to address the shortcomings of commonly used model order reduction and homogenization techniques. This dissertation specifically explores the feasibility and potential of fractional calculus to overcome some of the most significant limitations of discrete and continuous parameter methods with specific application to the vibration and wave propagation analysis of structural systems. From the perspective of discrete parameter models, model order reduction methodologies based on time fractional differential equations are presented. The use of a frequency-dependent fractional order is capable of simultaneously delivering high accuracy and high levels of reduction across a wide frequency spectrum. On the other hand, for the case of continuous parameter systems, the research explores how space fractional operators can lead to alternative forms of homogenization for partial order differential equations with application to wave propagation in heterogeneous structures. Specific applications to one-dimensional elastic metamaterials and structural components embedded with acoustic black holes are presented. The physical interpretation, potential, benefits, and even limitations of the developed fractional models are examined in-depth.

Degree

Ph.D.

Advisors

Semperlotti, Purdue University.

Subject Area

Acoustics|Systems science

COinS