I Forward to black-box software

This introductory chapter represents a theoretical analysis of the computational algorithms for a numerical solution of the basic equations in continuum mechanics. In this chapter the general requirements for computational grids, discretization, and iterative methods for black-box software are examined. Finally, a concept of a two-grid algorithm for (de)coupled solving multidimensional nonlinear (initial-)boundary value problems in continuum mechanics (multiphysics simulation) in complex domains is presented.

I.1 Formalization of scientific and engineering computations

In the early days of computers, every machine computation constituted a notable scientific event. Since the early 1980s, the development of high-speed digital computers has had a great impact on the activities of engineers, physicists, chemists, and other non-mathematicians solving time-consuming engineering and scientific problems. It immediately became clear that specialized software would significantly reduce the work of coding and debugging in scientific and engineering applications.

Several industries and engineering and consulting companies worldwide use commercially available general-purpose CFD codes for the simulation of fluid flow, heat, and mass transfer and combustion in aerospace applications (Fluent, Star-CCM+, COMSOL's CFD Module, Altair's AcuSolve, and others). Also, many universities and research institutes worldwide apply commercial codes, besides using those developed in-house. Today open-source codes such as OpenFOAM are also freely available. Other important issues are the description of complex domain geometries and the generation of suitable grids.

However, to successfully apply such codes and to interpret the computed results, it is necessary to understand the fundamental concepts of computational methods. A promising and challenging trend in numerical simulation and scientific computing is to devise a single code to handle all problems already solved.

As a rule, the mathematical modeling consists of the following stages:

1)

The formulation of the mathematical model for the studied physical and chemical processes in the form

(1.1)N(u)=f.

2)

The approximation of the space-time continuum (generation of a computational grid G).

3)

The approximation of the differential problems (1.1) on the grid G to obtain a discrete analogue of the mathematical model

(1.2)Nh(uh)=fh.

4)

A numerical solution for the (non)linear discrete equations

(1.3)uh=Nh-1fh

on a sequential or parallel computer.

5)

The visualization and analysis of computational results.

Here N(u)=f is a system of (non)linear (integro-)differential equations and (initial-)boundary conditions (mathematical model), Nh(uh)=fh is the resulting system of (non)linear algebraic equations (the discrete analogue of the mathematical model), and uh=Nh-1fh is the numerical solution.

Unfortunately, each stage of the mathematical modeling is a very complex problem, which has not yet been solved robustly. The most time-consuming step in execution is the numerical solution of (non)linear discrete equations (1.3).

The determining conditions for the software development are the wishes of potential users. However, what do the engineers, physicists, chemists, and other non-mathematicians want? Non-mathematicians want to operate with convenient and customary objects (domain geometry, initial and boundary conditions, working environments, source terms, fundamental equations, etc.) and get computational results in their obvious form, and they do not want to know anything about grids, approximations, linearizations, iterative methods, parallel computations, computer architectures, and other details of numerical experiments. Users need a very easy-to-use and powerful tool that will significantly expand their capabilities without the need to learn computational mathematics and programming, i.?e., black-box software. It is natural to want to minimize execution time for solving real-life problems using simple computers. Our definition of black-box software is given in the Preface.

All problems caused by developing black-box software for solving real-life applications can be subdivided into "physical", "mathematical", and "computer" subproblems:

• "Physical" subproblems arise from the difficulties of mathematically describing complex physical and chemical processes (1.1), such as hydrodynamics, heat and mass transfer in multiphase reacting flows, turbulent combustion, etc.

• "Mathematical'' subproblems arise from the formalization complexity of the main stages of computational experiments: generation of computational grids, approximations of the governing system of nonlinear (integro-)differential equations (1.1), and the efficient solution of systems of nonlinear algebraic equations (1.2) on a sequential or parallel computer (1.3).

• "Computer" problems arise from the compatibility difficulties of frequently updated software and hardware.

"Physical" subproblems are a consequence of the variety of modeled physical and chemical processes and accuracy required for their mathematical description, whereas the imperfection of numerical methods for solving the governing nonlinear PDEs leads to "mathematical" subproblems.

The idea behind robust multigrid algorithms is to choose the components independently of a given problem to match as large a class of problems as possible. The robust approach is often used in software packages where attaining the highest efficiency for a single problem is not so important [35]. Further, it is supposed that the number of problem-dependent components defines the robustness of the algorithm:

Definition I.1.

Let there be a set of algorithms for solving some problem. An algorithm from this set is called robust if it has the lowest number of problem-dependent components.

This book focuses on "mathematical'' subproblems. To overcome the problem of robustness, a two-grid algorithm is developed for black-box software, but nonlinear problems are not transferred between these grids. The basic solver consists of two ingredients, smoothing on the original grid and correction of the auxiliary grid. The main feature of the proposed approach is application of the Robust Multigrid Technique (RMT) for computing the correction on the auxiliary structured grid (Chapter II). The RMT is a single-grid (pseudo-multigrid) algorithm with the lowest number of problem-dependent components, close-to-optimal complexity, and high parallel efficiency. Our considerations for the development of black-box software are summarized in the last section.

I.2 Mathematical models in the continuum mechanics

Continuum mechanics, a large branch of mechanics, is devoted to the study of the motion of gaseous, liquid, and rigid deformable bodies. In continuum mechanics, with the aid of and on the basis of methods and observations developed in theoretical mechanics, motions of material bodies that fill the space in a continuous manner and the distances between the points of which change during motion are examined. Besides ordinary material bodies, such as water, air, or iron, special media are also investigated in continuum mechanics, i.?e., fields of electro-magnetism, radiation, gravitation, etc. [32].

These are the basic hypotheses of continuum mechanics:

1)

The concept of a continuous medium must first of all be defined. All bodies consist of individual particles; however, there are very many such particles in any volume that are essential for the ensuing studies, and it is possible to assume that a medium fills the space in a continuous manner. Water, air, iron, etc. will be considered to be bodies that completely fill a part of space. Such a hypothetical continuous matter is termed a continuum. Not only common material bodies can be considered continuous continua, but also fields, for instance, electromagnetic fields. Such an idealization is necessary to employ the apparatus of continuous functions, differential and integral calculus for studying the motion of deformable bodies [32].

2)

Only...