A brief review on finite element method

In this article, Jeff Zhang elaborates on his previous work: Computer Aided Engineering (CAE).
The following piece highlights the Finite element method (FEM). One of the most important engineering analysis techniques. 



Finite element method (FEM) is one of the most important engineering analysis techniques. It is widely used in elastoplastic mechanics, fracture mechanics, fluid mechanics, heat conduction and other fields.

The basic idea of FEM is to discretize the structure and use a finite number of simple elements to represent complex objects.

The elements are connected to each other through a finite number of nodes, and then they are comprehensively solved according to the equilibrium and deformation coordination conditions. Since the number of elements is limited, the number of nodes is also limited, it is called the Finite Element Method.


How it began

As a product of the computer age, FEM is a numerical calculation method developed since the 1960s.
The international community began to invest a lot of manpower and resources in the development of finite element analysis programs as early as the early 1960s, although the real CAE software was born in the early 1970s, and the past 15 years have been the development stage of CAE software commercialization. In order to meet market demand and be adapted to the rapid development of computer hardware and software technology, functions, performance, user interface, and pre- and post-processing capabilities FEM software products have been greatly improved and expanded.
This makes the well-known CAE software on the market meet the current needs of users in terms of function, performance, ease of use, reliability and adaptability to the operating environment, thereby helping users solve thousands of problems. Practical engineering problems have also made indelible contributions to the development of science and technology and engineering applications.


Current software applications

Currently, popular CAE analysis software mainly includes NASTRAN, ADINA, ANSYS, ABAQUS, MARC, MAGSOFT, COSMOS, etc. MSC-NASTRAN software has quite a popularity in the aerospace field because of its special relationship with NASA. It is based on the earliest linear finite element analysis system mainly used in aerospace and merged with PDA’s PATRAN.
DYTRAN was organized and developed on the basis of DYNA3D, which specializes in impact and contact applications.

Recently, the non-linear analysis software MARC has been merged to become the largest finite element analysis system in the world.
ANSYS software is dedicated to the analysis and calculation of coupled fields. It can calculate the four fields of structure, fluid, heat and electromagnetic, and has won the love of thousands of users in the world.
ABAQUS is well-known for nonlinearity solving capability, which covers material nonlinearity, geometric nonlinearity and state nonlinearity, etc. ABAQUS software has a faster convergence speed and is easier to operate and use. Now ABAQUS is one of the most important components within simulia chain which integrates fe-safe(durability analysis), Tosca(topology Optimization), isight(parametric Optimization).
ADINA was developed by MIT and Its single system can perform the coupled calculation of structure, fluid and heat. And it has both implicit and explicit time integration algorithms. Because of its powerful functions in nonlinear solving, fluid-structure coupling analysis, etc., it has quickly become a rising star in finite element analysis software and has become the preferred software for nonlinear analysis and calculation.

Finite Element Method

Finite Element Method

The development trend of a finite element

With the continuous expansion of the application of finite element technology, its development presents the following characteristics


1. Seamless integration with CAD software

Nowadays, one of the development trends of finite element analysis software is integration with general CAD software. Which means after the modelling design of components and parts is completed with CAD software, the model can be directly transferred to CAE software for finite element meshing and analysis.
If the result of the analysis does not meet the design requirements, a re-design is required and will be analysed until satisfaction, thereby greatly improving the design level and efficiency.
In order to meet the requirements of engineers to quickly solve complex engineering problems, many commercial finite element analysis software has developed interfaces with well-known CAD software (such as Pro/ENGINEER, Unigraphics, SolidEdge, SolidWorks, IDEAS, Bentley, AutoCAD, etc.).
Some CAE software adopts CAD modelling technology in order to realize seamless integration with CAD software. For example, ADINA software adopts Parasolid core-based solid modelling technology and can be combined with Parasolid-based CAD software (such as Unigraphics, SolidEdge). , SolidWorks) to achieve truly seamless two-way data exchange.


2. More powerful mesh pre-processing capabilities

The basic process of the finite element method mainly includes three parts:

  • The discretization of the analysis object;
  • The finite element solution;
  • the post-processing of the calculation results.

Since the quality of the mesh by which the structure discretized directly affects the solution time and the correctness of the solution results, software developers have increased their investment in mesh processing, in recent years, so that the quality and efficiency of mesh generation are both greatly improved, but some aspects have not been improved yet, such as automatic hexahedral meshing of the 3D solid model and adaptive meshing of the model according to the solution results.
Most analysis software still does not have this feature. Automatic hexahedral meshing refers to the three-dimensional solid model program that can automatically divide hexahedral mesh elements.
Now, most software can use mapping, dragging, sweeping and other functions to generate hexahedral elements, but these functions can only be used for simple models. For complex 3D models, only automatic tetrahedral meshing technology can be used to generate tetrahedral elements.
For tetrahedral elements, incorrect results will be produced without intermediate nodes in some cases, a series of problems in solving time and convergence speed will be caused with intermediate nodes. Therefore, industries urgently hope that automatic hexahedral meshes replacing tetrahedral elements.


The emergence of the mesh function

Adaptive meshing refers to a cyclic process of estimating calculation errors, remeshing and recalculating based on the existing meshes based on the results of finite element calculations. For many practical engineering problems, during the whole solution process, some areas of the model will produce a lot of strain, causing element distortion, which leads to the failure of the solution or incorrect solution results. Therefore, automatic remeshing must be performed. The adaptive mesh is often a necessary condition for large strain analysis of many engineering problems such as crack propagation and sheet forming.


3. From solving linear problems to solving nonlinear problems

With the development of science and technology, the linear theory is far from meeting the requirements of the design.
Many engineering problems such as material failure and failure, crack growth, etc. cannot be solved by linear theory alone.
Non-linear analysis, such as thin plate forming must be performed. It is required to consider the large displacement, large strain (geometric nonlinearity) and plasticity (material nonlinearity) of the structure at the same time; when analyzing plastics, rubber, ceramics, concrete and geotechnical materials or considering the plasticity and creep effects of materials.


The material nonlinearity must be considered

As we all know, the solution of nonlinear problems is very complicated. It not only involves many specialized mathematical problems but also must master certain theoretical knowledge and solving skills. It is also difficult to learn. For this reason, some companies have spent a lot of manpower and material resources to develop nonlinear solution analysis software, such as ADINA, ABAQUS, etc.
Their common feature is an efficient nonlinear solver, a rich and practical nonlinear material library, and ADINA also has both implicit and explicit time integration methods.


4. From the solution of the single structure field to the coupled field problem

FEM was first applied in the aerospace field, mainly used to solve linear structural problems. The practice has proved that this is a very effective numerical analysis method. Theoretically, it has been proved that as long as the unit used for the discrete solution object is small enough, the obtained solution can be close enough to the exact value. FEM software used to solve structural linear problems have been relatively mature, and the development direction is the solution of structural nonlinearity, fluid dynamics and coupled field problems. For example, the thermal problems caused by frictional contact and the thermal problems caused by plastic work during metal forming require the cross-iterative solution of the finite element analysis results of the structure field and the temperature field, that is, the problem of “thermo-mechanical coupling”. When the fluid flows in the elbow, the fluid pressure will deform the elbow, and the deformation of the tube, in turn, affects the flow of the fluid. This requires the cross-iterative solution of the finite element analysis results of the structure field and the flow field. This is the so-called “fluid-structure coupling” problem.
As the application of finite element becomes more and more in-depth and people’s concerns become more and more complex, the solution of coupled fields must become the development direction of CAE software.


5. Openness of users programming

With the improvement of commercialization, in order to expand market share and meet the needs of users, software developers have spent a lot of investment in software functions and ease of use. However, due to the wide range of user requirements, no matter how hard they work It is impossible to meet the requirements of all users. Therefore, users must be given an open environment to allow users to expand the software according to their actual conditions, including user-defined unit characteristics, user-defined material constitutive (structure constitutive, thermal Structure, fluid constitutive), user-defined flow field boundary conditions, user-defined structural fracture criterion and crack growth law, etc.



With the development of computer technology, FEM has been continuously applied in various engineering fields. It has now spread to the aerospace industry, nuclear industry, electromechanical, chemical, construction, marine and other industries. It is the analysis of dynamic, static and thermal characteristics of mechanical products.

At present, the finite element method is still being developed and improved theoretically. Various finite element analysis packages are becoming more powerful and more convenient to use.

About the authors

David de Brouwer

Founder / Managing Director Engibex

Engibex consultant - Jeff Zhang

Jeff Zhang

Doctor of Philosophy (PhD), Electrical Mechanics Engineering

Expertise in numerical modelling & experimental data mining. Great enthusiasm in innovative product design & R&D.