Finite Element Analysis Shoots for the Cloud

History of Simulation

June 8, 2021

Engineers and scientists often mistakenly associate Finite Element Modeling with numerical simulation. It is an honest assumption given the fact that both finite element modeling and numerical simulation aim to numerically approximate the solution to a physical problem in a complex domain. The reality though is that it was not until the 1980s that numerical simulation was weaved into FEA technology. Furthermore, while it is currently routinely taught in graduate schools across the world today, finite element modeling was first used in the 1950s as a replacement for the traditional force method. 

Part 1 of our History of Simulation series, A History of Finite Element Analysis: From Numerical Methods to Cloud-based Simulations, focused on numerical analysis and numerical methods, highlighting different classes, or numerical methods, as a precursor for this blog. This article, part 2 of the series, begins with the history of finite element methods and the people involved in making this mathematical wonder a reality and ends with a description of the three main FEM competitors: finite volume, boundary element, and meshless methods.

Below is a chronological timeline of the who, when, and how of the development of FEM

  • FEM is being developed at Boeing
  • Not one individual can be credited to be the inventor of finite elements, however, M. J. Turner at Boeing generalized and perfected the Direct Stiffness Method and convinced Boeing to move away from the Force Method and embrace finite elements. 
  • Turner’s article on the finite element method was published by Turner, R. W. Clough, H. C. Martin, and L. J. Topp in 1956, and is widely recognized as the first paper in this new computational method. Download a PDF of the article, Stiffness and Deflection Analysis of Complex Structures, here.
  • Clough allegedly coined the term “finite elements” a few years later in 1960.
  • As the popularity of the method grew, in 1965 NASA issued a request for proposal for the development of structural analysis software. NASTRAN (NASA Structural Analysis) was developed by Computer Sciences Corporation who used the existing FEM technology to solve structural problems.
  • In the 19th and 20th centuries, popularization of the finite element method came from the early pioneers such as Dr. O.C Zienkiewicz, who is credited with writing the first book purely focused on FEM titled “The Finite Element Method” published in 1967.
  • In 1968 I. Ergatoudis, B. M. Irons, and Zienkiewicz introduced ‘isoparametric’ mapping for quadrilateral elements. 
  • This is also the year that NASTRAN development is completed, and the software was first commercialized by MacNeal-Schwendler Corporation in 1969. This begins the FEM “golden age” that will last until roughly 1972.
  • During this period higher-order displacement elements are developed to address the disappointing performance of linear elements.
  • Finite Volume (FV) is introduced to the CFD field.
  • FEM is mathematically proven in the so-called “consolidation period” that lasted until approximately 1980. 
  • The first mathematical proofs on the properties of the FEM; convergence and uniqueness of the solution, were published in 1972.
  • Smoothed Particle Hydrodynamics (meshless methods) is born.
  • FIESTA, the first professional p-version code for FEM, was initiated by Dr. Alberto Peano in 1977. The initial application for this code was to perform numerical analysis of dams and other civil engineering structures.
  • The period between 1980 and 1990 saw the rise of the p-version FEM after the publication of p-version convergence in 1981 by Babushka, Szabo, and Katz.  A flurry of development activities and acquisitions followed, with FIESTA, PROBE, and MECHANICA being the most notable FEM software developed during this period. 
  • At the same time, Babuska and Szabo provided the first proof of an exponential rate of convergence using the p-version of the FEM; convergence was assumed to be linear in a log-log scale as the number of elements was increased up to 1984.
  • The Upperboard method is implemented in DEFORM software.
  • After the 1990s many commercially available FEM software became available to academia and industry users. Ansys, Comsol, Abaqus, Pro-Mechanica, BEASY, OpenFOAM, and many more software packages directed their efforts to parallel processing and cloud computing. 
  • After the 1990s many commercially available FEM software became available to academia and industry users. Ansys, Comsol, Abaqus, Pro-Mechanica, BEASY, OpenFOAM, and many more software packages directed their efforts to parallel processing and cloud computing. 
  • The era of parallel processing and cloud computing begins.

As finite element garnered traction in solid mechanics and potential field problems such as heat transfer and computational fluid dynamics, there were many spin-off formulations that were developed over the years including:

  • Finite Volume Method for Computational Fluid Dynamics
  • Boundary Element Method for infinite media
  • Upper-bound methods for metal processing (forging in particular)
  • Meshless methods that do not use the conventional grid approach of FEM but a point-cloud approach

We will now take a closer look at several methods, emphasizing the differences between methods, and describe their principle of operations. 

Finite Volume

Finite Volume is a popular method used in the field of Computational Fluid Dynamics (CFD). Finite Volume Method (FVM) evaluates exact expressions for the average value of the solution over some volume (therefore the name finite volume) and uses these values to construct approximations of the solution within the discretized cells.  This is different from the Finite Difference Method (FDM) which approximates derivatives using nodal values, or the Finite Element Method (FEM) which approximates the solution locally at the element level using local values and assembles the global solution by stitching together the local approximations. 

You can think of FVM as a method that decomposes the domain of interest using a fixed “volumetric” mesh and lets the fluid pass through the mesh as it computes the scalar variables of interest such as speed, temperature, pressure (Eulerian simulations). This is somewhat different from structural mechanics FEM that computes material derivatives and displacements on deformable elements and updates the solution as the load is gradually increased to the final value (Lagrangian simulations). 

the difference between Lagrangian vs Eulerian formulation
This image shows the difference between Lagrangian vs Eulerian formulation

Boundary Element Method

Boundary Element Method (BEM) is a numerical computational method of solving linear PDEs which have been formulated in boundary integral form. BEM solves boundary value problems (BVP) by trying to use the given boundary condition to fit boundary values in the integral equation and compute numerically the solution directly at any desired point in the interior of the solution domain. Besides the reduced size of the models (only surface meshes are needed for BEM), the method is very suitable for solving infinite or semi-infinite problems such as CFD over an aircraft or soil mechanics.

Meshless methods

These methods discretize the domain in a node cluster that does not require connections between nodes (a mesh). The methods use interactions between neighboring nodes rather than mesh elements. Meshless methods enable modeling and simulation of difficult problems such as crack propagation, soil mechanics, at the expense of computing time and complex mathematical implementation. The absence of mesh also allows meshless methods to use the Lagrangian formulation in their implementations,  similar to the FEM analysis approach.

While we have not exhausted the topic of FEM, the articles in this series tell the story of the what, when, and who were involved in finite element analysis from its inception. The roots and the history of this fascinating simulation technology are rich and fascinating. 

The finite element method has come a long way. There are many companies today that commercialize FEM software both as standalone and flexible, cloud-based solutions. The most notable recent developments in the FEM world take advantage of the leaps in computational technology and hardware. Numerical algorithms have to be optimized to take advantage of shared-memory parallel computers and run efficiently in the cloud.  Artificial intelligence and genetic algorithms are likely one of the next steps for cloud-based FEM analysis. The future of computer modeling and simulation has just begun.

History of Simulation series
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