In this blog post, we’ll explore the principles, applications, and limitations of finite element analysis—a method that breaks down complex structures, like LEGO sets, into small elements for analysis.
Among the countless toys loved by children, LEGO is a must-mention. LEGO is a toy consisting of colorful blocks of various sizes that are interlocked to build large objects; its appeal lies in the endless possibilities for shapes that can be created depending on the size of the blocks used and the assembly method. Interestingly, a similar concept is also put to good use in cutting-edge science. Let’s find out how it’s applied.
Usually, when you take the subway across the Han River and look out the window, you see the beautiful river and the impressive buildings surrounding it. However, sometimes the first thing that catches your eye is the dust-covered steel structure of the bridge rushing past. You can easily see thick steel rods connected in all sorts of ways. Looking at it, you might wonder: “Why did they connect them in such a complicated way when it could have been made simpler?” Or, “Those bars look kind of unstable—if they break, won’t the bridge collapse? After all, the Seongsu Bridge did collapse in the past…” However, these concerns can be alleviated to some extent by understanding the process of evaluating a structure’s design and safety. The stability of bridges and buildings is evaluated during the design process by considering various loads and conditions, and standards and safety factors are applied to ensure the structure’s safety. So, how can we know whether a bridge will collapse or not even before it is built? This is where structural analysis comes in, and one of the most widely used methods is finite element analysis. Structural analysis is a crucial element in research and experimentation at the School of Mechanical and Aerospace Engineering. Such structural analysis plays a vital role not only in the Department of Mechanical and Aerospace Engineering but also in various other engineering fields, as well as in architecture and manufacturing. Structural analysis involves analyzing the deformation or behavior of an object based on conditions such as its shape, material, applied loads, and vibrations. Simply put, it determines whether a given object or structure can withstand a specific environment without breaking or deforming excessively.
For simple objects—such as straight rods, square bricks, or circular pipes—structural analysis is not difficult and can be solved relatively simply using mathematics and mechanics. However, in most real-world applications requiring structural analysis, the structures involved are highly complex. Buildings, spires, automobiles, and airplanes consist of a large number of components and are often quite large, making it difficult to analyze their entire structure by directly applying simple mechanical formulas. Furthermore, since the applied loads and external environments are highly diverse and complex, analyzing the stability of such structures becomes even more challenging. This is where finite element analysis comes into play. The Finite Element Method (FEM), as the name suggests, is a method that divides a complex problem into a finite number of simple elements, analyzes each element individually, and then combines the results. In other words, by viewing a complex object as a collection of simple elements, it becomes possible to analyze each element individually, thereby enabling the calculation of the overall behavior of the complex object. This process is called discretization or mesh generation. The concept is similar to how simple, brick-shaped LEGO blocks are combined to create objects with complex shapes. However, unlike LEGO blocks, which are physically separate objects, finite elements are not discrete objects; rather, they represent a continuous structure divided into small regions for computational purposes. Each element is connected to others via nodes, and the behavior of the original structure is approximated using the overall mesh formed by these elements. This is the core of the finite element method.
The size, shape, and material properties of these elements can be configured to suit the characteristics of the problem. As one might intuitively expect, increasing the number of elements that make up the object appropriately and reducing the size of elements in necessary regions generally leads to more precise analysis results. If the number of elements is too small, the changes in the structure’s geometry or physical quantities may not be adequately represented; however, by appropriately subdividing the elements, the error between the original geometry and the computational model can be reduced. That said, making the elements smaller indiscriminately is not always beneficial. Since an excessive number of elements increases computational load and time, it is important to create a fine-grained mesh focused on regions of interest or areas where stress concentrations are expected. In modern finite element analysis, computers perform these numerous equations and matrix calculations. The finite element method began to develop in earnest in the 1940s and advanced rapidly in the field of engineering in the 1950s as it was applied to the analysis of aircraft structures and other applications. Subsequently, as computer performance improved significantly, it became possible to process large-scale finite element models that were previously difficult to compute; today, large-scale analyses using tens of thousands to millions or more elements are performed depending on the field and problem.
The bridges we cross every day may appear, from the outside, to be constructed from several connected steel members, but during the actual design process, the structure can be divided into countless elements for analysis. If there is a possibility that the structure could deform or be damaged due to large forces, it is possible to analyze where high stresses occur, which parts exhibit significant deformation, and which areas need reinforcement to prevent member failure. This finite element analysis is utilized not only for mechanical structural analysis but also for a wide range of other problems. For example, it can be used to analyze the amount and path of heat transfer, predict the flow of fluids such as water or air, and analyze spaces where electric and magnetic fields are formed. Furthermore, in the field of biomedical engineering, it can be used to analyze the behavior of structures such as the spine, skull, pelvic joints, teeth, and the heart. Finite element analysis can also be utilized in the design and evaluation of artificial teeth, dental prostheses, and artificial bones used in dentistry. As such, the finite element method is not limited to structural analysis but is applied to numerical analysis problems in various fields, including heat transfer, fluid dynamics, electromagnetic fields, and biomedical engineering, and has established itself as a key method for analyzing complex real-world problems using computers.
However, finite element analysis also has its limitations. Although advances in computer technology have made it possible to perform a vast number of calculations at high speeds, significant computational resources are still required for complex nonlinear problems or large-scale coupled fluid-thermal-structural problems. In particular, a high-performance computing environment may be required when calculating complex phenomena that change over time—such as fluid flow problems—with a high degree of precision. Furthermore, it cannot be assumed that the results of computer simulations will appear exactly the same in actual structures. This is because actual structures involve various variables that are difficult to incorporate into a model; factors such as material properties, errors arising during the manufacturing process, differences between actual loads and boundary conditions, and changes in the surrounding environment can all influence the results. Therefore, rather than blindly trusting computer simulation results, it is essential to verify how well the assumptions and input conditions used in the analysis reflect the actual situation and, if necessary, validate the results through experiments or other analytical methods. This is why verifying calculation results and comparing them with actual behavior is crucial in finite element analysis as well.
For this reason, when designing actual structures, sufficient margins are incorporated to account for anticipated loads and various uncertainties, ensuring the structure reliably meets the required performance standards. The primary concept representing the margin of safety a structure has against design loads is the safety factor. The safety factor is determined based on the type of structure, design standards, materials, load conditions, and failure modes; therefore, the same value is not applied to all structures. Finite element analysis is also a very useful tool for assessing structural safety, but it requires a process to account for differences between the analysis model and the actual structure and to properly validate the results. Therefore, developing more precise modeling and efficient calculation methods, and improving the reliability of finite element analysis results in relation to the actual behavior of structures, remains an important challenge.
