In structural analysis software like RFEM, the term "integration" often refers to the numerical integration process used to solve differential equations that arise from the finite element analysis. This process is crucial for determining how the structure responds to applied loads and boundary conditions. Here is a simplified overview of the mathematical integration process in the context of finite element analysis:
- Discretization: The continuous physical behavior of a structure is represented by a set of differential equations that describe how forces, stresses, displacements, and other parameters are related. These equations are typically partial differential equations (PDEs). To solve those equations numerically, the first step is to discretize the problem by dividing the structure into smaller elements (such as triangles or tetrahedra for 2D or 3D analyses).
- Local Equations: Within each element, the equations describing the behavior of the structure are formulated. These equations relate the local displacements, strains, and stresses within the element.
- Gaussian Quadrature: The process of numerical integration is often performed using Gaussian quadrature. This method approximates the integral of a function by evaluating the function at a set of discrete points within the element and then combining those evaluations using specific weights.
- Assembly: The global behavior of the entire structure is determined by combining the local behaviors of each element. This is achieved through the assembly process, where the contributions of neighboring elements are combined to form the overall system of equations.
- Boundary Conditions: The boundary conditions, such as fixed supports or applied loads, are applied to the assembled system of equations. This involves modifying the equations to account for the constraints and forces applied to the structure.
- Solution: The modified system of equations is solved to determine the unknown displacements and other response parameters. This solution involves solving a large system of linear equations, which can be done using various numerical methods, such as direct solvers or iterative techniques.
- Post-Processing: Once the displacements and other response parameters are obtained, post-processing is performed to calculate additional results – stresses, strains, reactions, and displacements at specific locations of interest in the structure. Those results help engineers assess the structural performance and ensure it meets design requirements.
- Iterative Process: The process might involve iterating through steps 1 to 7 to refine the analysis, adjust input parameters, or investigate different scenarios until a satisfactory solution is obtained.