By now, it is abundantly clear that all computational aeroacoustics (CAA) simulations can only be carried out in a finite, and in most cases, as small as possible, computational domain. A critical problem is how to continue the numerical solution to the far field. In some cases, the acoustic field consists of discrete frequency sound or tones. But in most cases, such as jet noise, the sound field is broadband and random. In aeroacoustics, most broadband noise is the result of turbulence. Turbulence is random and definitely nondeterministic, and the same is true of the radiated noise. The nondeterministic nature of the noise of turbulence makes its continuation to the far field a much more challenging task.

Mathematically, the problem of extending a near-field acoustic solution to the far field is akin to the mathematical procedure of “analytic continuation” in complex variables. Analytic continuation extends an analytic function defined in a limited domain to a large domain. Because the ideas behind the two problems are similar, the word “continuation” is used here to indicate the objective and intent.

At high Reynolds numbers, a flow is inevitably turbulent. As noted earlier, a turbulent flow is chaotic and random. For a turbulent flow such as that of a high Reynolds number jet, the solution is nonunique. Nonuniqueness is a characteristic of turbulence. Only the statistical averaged quantities are stationary with respect to time. This being the case, it brings forth the question of what physical variables should be continued to the far field. There is also the question of cost in performing the continuation. One may wish to continue the entire turbulence information to the far field, but the computational cost is likely to be prohibitive. Thus, this may not be a good idea. From a practical standpoint, the crucial information one needs in the far field is the noise directivity and spectra. If, indeed, the noise spectra and directivity are the information required in the far field, then a not so expensive computational procedure may be developed to continue this information from the near to the far field. In this chapter only the continuation of the spectra and directivity of turbulence noise are considered.

In this chapter we present notation and preliminary material which is necessary in the study of the boundary value problems presented in Chapters 5–7. We start by introducing some function spaces that will be relevant in the study of contact problems. Then we provide a general description of the mathematical modelling of the processes involved in contact between an elastic, viscoelastic or viscoplastic body and an obstacle, say a foundation. We describe the physical setting, the variables which determine the state of the system, the material's behavior which is reflected in the constitutive law, the input data, the equation of equilibrium for the state of the system and the boundary conditions for the system variables. Finally, we extend this description to the contact of piezoelectric bodies for which we consider both the case when the foundation is conductive and the case when it is an insulator.

Everywhere in the rest of the book we assume that Ω ⊂ ℝd (d = 1, 2, 3) is open, connected, bounded, and has a Lipschitz continuous boundary Γ. We denote by = Ω ∪ Γ the closure of Ω in ℝd. We use bold face letters for vectors and tensors, such us the outward unit normal on Γ, denoted by ν. A typical point in ℝd is denoted by x = (xi). The indices i, j, k, l run between 1 and d, and, unless stated otherwise, the summation convention over repeated indices is used.

In this chapter we illustrate the use of the abstract results obtained in Chapter 3 in the study of quasistatic frictionless and frictional contact problems. We model the material's behavior either with a nonlinear viscoelastic constitutive law with short or long memory, or with a viscoplastic constitutive law. The contact is either bilateral or modelled with the normal compliance condition with or without unilateral constraint. The friction is modelled with Coulomb's law and its versions. For each problem we provide a variational formulation which is in the form of a nonlinear equation or variational inequality for the displacement or the velocity field. Then we use the abstract existence and uniqueness results presented in Chapter 3 to prove the unique weak solvability of the corresponding contact problems. For some of the problems we also provide convergence results. Everywhere in this chapter we use the function spaces introduced in Section 4.1 and the equations and boundary conditions described in Section 4.4.

Bilateral frictionless contact problems

In this section we study two frictionless contact problems for viscoelastic materials in which the contact is bilateral. The main feature of these problems consists of the fact that their variational formulation is in the form of a nonlinear equation for the displacement field, which is either evolutionary or involves a history-dependent operator. Therefore, the unique solvability of the models is proved by using the abstract results presented in Section 3.1.

In this chapter we deal with the study of variational inequalities which involve the so-called history-dependent operators. We start with some preliminary material on spaces of vector-valued functions. Then, in order to illustrate the main ideas which are developed in this chapter, we analyze two nonlinear equations which lead to history-dependent operators. We proceed by introducing the concept of the history-dependent quasivariational inequalities together with a theorem on their unique solvability. Also, we state and prove a general convergence result in the study of these inequalities. Next, we particularize our results in the study of evolutionary variational and quasivariational inequalities. Besides their own interest, the results presented in this chapter represent crucial tools in deriving the existence of a unique weak solution to frictionless and frictional contact problems with viscoelastic and viscoplastic materials. Even if most of the results we present in this chapter still remain valid for more general cases, we restrict ourselves to the framework of strongly monotone Lipschitz continuous operators in Hilbert spaces as this is sufficient for the applications we consider in Chapter 6. Everywhere in this chapter X denotes a real Hilbert space with inner product (·,·) X and norm ⊂ ·⊂X.

Nonlinear equations with history-dependent operators

In this section we consider two nonlinear equations in Hilbert spaces which lead to history-dependent operators. This will allow us to illustrate the main idea of this chapter which consists of unifying the study of various types of evolutionary equations or inequalities, by using a convenient choice of the variables and operators.