Introduction

The main goal of the book is to establish a synthesis between classical matrix and tensor methods, and modern software technology. To achieve this, the approach involves the development of a cohesive methodological framework using object-oriented methods. This framework enables the seamless transformation of the theoretical modeling principles of mechanics into numerical computational programs for solving engineering problems. This is done without encountering methodological inconsistencies in various subareas. One classic example of such a challenge is the symbolic notation of tensor and matrix calculus, which features complex syntax and semantics. This notation, originally designed for 2D media, is still widely used today, especially in the form of matrix arithmetic. Despite the availability of multidimensional arrays in programming languages for matrix implementation, this notation remains prevalent.Consequently, the well-established index notation for tensors and matrices becomes invaluable when specifying array objects within matrix operations. It may appear less popular due to the extensive use of sub- and super-indices. However, when applied with Einstein’s summation convention, it becomes highly effective and allows for transparent handling of multidimensional matrix operations, akin to ordinary scalar objects. Therefore, the introductory chapters explicitly introduce both symbolic and index notations to get readers acquainted with both notations in detail.Index notation has grown immensely important for implementing tensor and matrix methods in object-oriented software designs. Multidimensional objects can be effectively implemented using specialized class structures in programming languages like C++ or Java. Compared to algorithmic programming languages such as C or FORTRAN, the object-oriented paradigm offers several advantages in declaring and defining tensor and matrix objects, as detailed by [Breymann 1997]. Some of the key topics covered in the book include:- Declaration and definition of new class members in structured and related classes,- Encapsulation of class member data and functions in private or protected areas (hidden and not accessible to the user),- Definition of public interfaces for communication with class members (e.g., by calling functions),- Inheritance of properties between class relations,- Dynamic generation and deletion of new class instances during run time, and additionally, C++ allows for:- Overloading of arithmetic and functional operators as per its language specifications.Using these features, matrix operations can be implemented similarly to familiar scalar operations. These aspects of the approach are described and utilized in Chapter 3, considering the relevant properties of matrices and tensors outlined in Chapters 2 and 4. Finally, the practical application of the synthesized tensor/matrix methods is demonstrated through two typical software examples for Finite Element Method (FEM) applications in Chapter 7.The primary objective of the book adheres to the rigorous approach that tensors, widely used in the context of consistent theories of mechanics and physics, should also be effectively utilized for numerical computations through object-oriented matrix methods in engineering. As such, it assumes a strong foundation in matrix and software methods, which are discussed in subsequent chapters.