The basic idea of the Finite Element Method is to discretize the structure, use a finite number of simple units to represent complex objects, and the units are connected to each other through a finite number of nodes, and then comprehensively solve according to the equilibrium and deformation coordination conditions. Since the number of units is limited and the number of nodes is also limited, it is called the Finite Element Method (FEM, Finite Element Method).
The finite element method is one of the most effective numerical calculation methods so far, and it provides great support for science and engineering technology.
The breeding process, birth and development of finite element method
In the 17th century, Newton and Leibniz invented the method of integration, proving that the operation has global-to-part additivity.
In the 18th century, the famous mathematician Gauss proposed the weighted residual method and the solution of linear algebraic equations. Another mathematician, Lagrange, proposed functional analysis. Functional analysis is another way to rewrite partial differential equations as integral expressions.
At the end of the 19th century and the beginning of the 20th century, mathematicians Rayleigh and Ritz first proposed that the displacement function can be used to express the unknown function on the full domain.
In 1915, the mathematician Galerkin proposed the Galerkin method to select the shape function in the displacement function, which is widely used in finite elements.
In 1943, the mathematician Courant proposed for the first time that the displacement function can be used to express the unknown function in the domain of definition. This is actually what finite elements do.
In the 1950s, aircraft designers found that it was impossible to use traditional mechanical methods to analyze the stress and strain of aircraft. A technical team of Boeing first discretized the wing of the continuum into a collection of triangular plates for stress analysis, and succeeded after some twists and turns. (Professor Clough participated in the research.)
In the 1950s, large-scale electronic computers were put into the work of solving large-scale algebraic equations, which prepared the material conditions for the realization of finite element technology.
In 1960, Professor R.W.Clough of the University of California, Berkeley proposed the term “finite element” in his paper. What is worthy of pride is that Professor Feng Kang of Nanjing University in my country independently proposed “finite elements” in his papers before and after.
Calculation method and software of finite element method
As a technology, the finite element calculation method is closely combined with the development of FEM software. The method is constantly updated, survival of the fittest, inheritance and development. Among the numerical calculation methods of traditional finite element analysis, there are direct calculation method (DirectSolver) and iterative method (Iterative so-called fast solution method).
Common finite element software includes: ABQUS, ADINA, ANSYS, MARC, COSMOS, ELAS, MSC and STARDYNE in the United States, ASKA in Germany, PAFEC in the United Kingdom, and SYSTUS in France, etc.
Explicit/implicit finite element method: only need to invert the mass matrix that can be simplified to a diagonal matrix, there is no iterative convergence problem in incremental steps, and the calculation can be continued. Implicit calculation has the characteristics of large time step increment, control of convergence at each load step, avoiding error accumulation, problem of non-convergence of iterations, and super-linear growth of calculation amount with the increase of calculation scale. Relative and implicit explicit calculations have the characteristics of small time step size, error accumulation, no iterative non-convergence problem, and the calculation amount basically increases linearly with the calculation scale. The representative software of this calculation method is ABQUS.
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Discrete element method: The discrete element method is also called the discrete element method. It was first proposed by Cundall in 1971 as a discontinuous numerical method model. The advantage of this method is that it is suitable for simulating discrete particle assemblies in quasi-static or dynamic deformation process under conditions. The discrete element method is not based on the principle of minimum potential energy variation, but on the most basic Newton’s second law of motion. It is based on the motion equations of each rigid body, and after establishing an explicit equation system describing the entire failure process, iteratively solves it through dynamic relaxation.
Rigid Body Spring Element Method: Rigid Body Spring Method (RBSM) was first proposed by Kawai in 1976. The original intention was to solve structural problems with fewer degrees of freedom. It decomposes the system into some rigid elements that are connected by spring systems evenly distributed on the contact surface. The rigid elements themselves do not undergo elastic deformation, so the deformation energy of the structure can only be stored in the spring system on the contact surface. Since the force between the rigid spring element elements is transmitted through the spring on the element interface, the interface force can be obtained directly, so it is also widely used in the field of rock-soil interface analysis and other fields.
Contact judgment method: through the mutual contact judgment between units, the mutual force is obtained, and then the motion equation is formed. Therefore, a fast and accurate contact algorithm is very important for the finite element method. Due to the relatively large displacement of the units during the calculation process, the spatial topological relationship between the original blocks changes, which makes the contact judgment more complicated.
Meshless method: The traditional finite element needs to construct a specific unit grid to form the position interpolation function. Is it possible to let the computer “automatically” form the displacement interpolation function according to the node information? Meshless method can be achieved. The requirements for the function of the mesh-free method are:
smooth and continuous;
Affected nodes are limited.
Commonly used interpolation methods for gridless methods are: moving least squares, kernel function and radial basis function. The overall equation has collocation method, least square method and Galerkin method. The Galerkin method is one of the most widely used and most stable mesh-free methods.
XFEM: Proposed in 1999, the extended finite element method (XFEM) has been greatly developed under the efforts of Belytschko and other scholars, and it has been realized in the version 6.10 software of ABAQUS.
The development trend of finite element method in the field of structural engineering
Multiphysics coupling problem
In recent years, the finite element method has been developed to solve the problems of fluid mechanics, temperature field, electric conduction, magnetic field, seepage and sound field, etc., and recently it has been developed to solve several interdisciplinary problems. If it is necessary to use the finite element analysis results of solid mechanics and fluid dynamics to solve iteratively, it is the so-called “fluid-solid coupling” problem.
From linear engineering problems to nonlinear analysis problems
Linear theory has been far from meeting the design requirements. For example, the emergence of high-rise buildings and long-span suspension bridges in civil engineering requires the consideration of geometric nonlinear problems such as large displacement and large strain of the structure;
There are thermal deformation and thermal stress in high-temperature components of aerospace and power engineering, and nonlinear problems of materials must also be considered; the emergence of various new materials such as plastics, rubber and composite materials, linear calculation theory alone is not enough to solve the problems encountered The problem can only be solved by using nonlinear finite element algorithm.
Time-varying structures and progressive collapse problems
Structures cannot exist there naturally, nor can they disappear out of thin air, so the construction or demolition process of structures is dynamic, and may show different mechanical properties at different stages, and there are many complicated problems in it. For this kind of process analysis, the finite element often compiles time-schedule programs according to the construction process, and dynamically tracks changes in structural performance.
Optimization problem:
There are several requirements in finite element, such as boundary shape optimization, minimum mass, equal strength, equal strain, dynamic parameter optimization, etc. The optimization problem is characterized by many variables (tens/hundreds), and many practical The robustness of optimization algorithms with so many variables needs to be improved.
Turbulence problem:
At present, there are some good methods, such as finite volume method, etc., which still need to be deepened. The current turbulence problem is not an algorithm problem at all, but a physical model problem of the medium; people’s understanding of turbulence may still be limited by the current level of science and technology. limit.
V. Conclusion
The finite element method is not omnipotent, the key lies in its thought, it perfectly embodies the relationship between the part and the whole in philosophy, to solve the overall problem, we must first study the local problem, after the local problem is clearly studied, then study the relationship between the parts , and then each part is synthesized under a unified coordinate scale, considering the relationship between the whole system and the outside, and finally the global features are obtained.
The finite element method is a scientific tool for us to understand the world, but its philosophical meaning and methodological meaning are far from being recognized by people.