In FEM, finite elements play a central role. The following chapters provide in-depth insights into various aspects of FEM, starting with the various types of finite elements, followed by a detailed description of the numerical integration process, the mesh settings optimization, and how to prevent zero-energy modes. Each chapter is designed to explain essential knowledge and techniques for effectively utilizing FEM in engineering projects.
Finite elements are essential for analyses according to the finite element method (FEM) – a numerical method widely employed in engineering and scientific disciplines to analyze and solve complex structural and physical problems. Here is a brief overview of finite elements in FEM:
- Definition: Finite elements are discrete, geometric subregions or elements into which a complex structure or system is divided. Those elements approximate the behavior of the entire structure, allowing for the analysis of complex systems through simpler, manageable components.
- Purpose: The primary purpose of dividing a complex domain into finite elements is to simplify the problem-solving process. Instead of solving a set of intricate differential equations over the entire domain, FEM breaks the problem down into a series of smaller, interconnected elements, making it computationally feasible.
- Interpolation: Within each finite element, interpolation functions – often based on polynomial approximations – are used to represent the variations in physical properties, such as displacement, stress, temperature, etc., across the element. Those functions allow engineers to approximate the continuous behavior of the structure.
- Assembly: In FEM analysis, the individual finite elements are assembled to form a global system of equations. The boundary conditions and applied loads are imposed within this global system, creating a representation of the entire problem.
- Solving Equations: The global system of equations is typically solved using numerical techniques, such as matrix algebra, iterative solvers, or direct methods, depending on the size and complexity of the problem.
- Results: Once the equations are solved, engineers obtain valuable information about the behavior of the structure, including displacements, stresses, strains, and other relevant physical quantities. The results have an impact on design decisions, optimizations, and evaluations.
- Applications: Finite elements find application in a wide range of engineering and scientific disciplines, including civil and structural engineering, mechanical engineering, aerospace engineering, and material science. They are instrumental in analyzing and designing structures, components, and systems subjected to various physical forces and conditions.
- Advantages: The advantages of using finite elements in FEM calculations include the ability to model complex geometries, heterogeneous materials, and nonlinear behaviors accurately. This offers a systematic approach to solving engineering problems and allows for the exploration of multiple design scenarios.
In summary, finite elements serve as the fundamental building blocks in finite element analysis, enabling engineers and scientists to simulate and analyze complex systems with precision and efficiency.