By Dr. Barna Szabó
Engineering Software Research and Development, Inc.
St. Louis, Missouri USA

In Thomas Kuhn’s terminology, “pre-science“ refers to a period of early development in a field of research [1]. During this period, there is no established explanatory framework (paradigm) mature enough to solve the main problems. In the case of the finite element method (FEM), the period of pre-science started when reference [2] was published in 1956 and ended in the early 1970s when scientific investigation began in the applied mathematics community. The publication of lectures at the University of Maryland [3] and the first mathematical book on FEM [4] marked the transition to what Kuhn termed “normal science”.

Two Views

Engineers view FEM as an intuitive modeling tool, whereas mathematicians see it as a method for approximating the solutions of partial differential equations cast in variational form. On the engineering side, the emphasis is on implementation and applications, while mathematicians are concerned with clarifying the conditions for stability and consistency, establishing error estimates, and formulating extraction procedures for various quantities of interest. 

From the beginning, a significant communication gap existed between the engineering and mathematical communities. Engineers did not understand why mathematicians would worry so much about the number of square-integrable derivatives, and mathematicians did not understand how it is possible that engineers can find useful solutions even when the rules of variational calculus are violated. This gap widened over the years: On one hand, the art of finite element modeling became an integral part of engineering practice. On the other hand, the science of finite element analysis became an established branch of applied mathematics.

The Art of Finite Element Modeling

The art of finite element modeling has its roots in the pre-science period of finite element analysis when engineers sought to extend the matrix methods of structural analysis, developed for trusses and frames, to complex structures such as plates, shells, and solids. The major finite element modeling software products in use today, such as NASTRAN, ANSYS, MARC, and Abaqus are all based on the understanding of the finite element method (FEM) that existed before 1970. As long as the goal is to find force-displacement relationships, such as in load models of airframes and crash dynamics models of automobiles, finite element modeling can provide useful information. However, problems arise when the quantities of interest include (or depend on) the pointwise derivatives of the solution, as in strength analysis where stresses and strains are of interest.

Misplaced Accusations

The first mathematical book on the finite element method [4] dedicated a chapter to violations of the rules of variational calculus in various implementations of the finite element method. The title of the chapter is “Variational Crimes,” a catchphrase that quickly caught on. The variational crimes are charged as follows:

1. Using non-conforming Elements: Non-conforming elements are those that do not satisfy the interelement continuity requirements of the variational formulation.
2. Using numerical integration.
3. Approximating domains and boundary conditions.

Item 1 is a serious crime, however, the motivations for committing this crime can be negated by properly formulating mathematical models. Items 2 and 3 are not crimes; they are essential features of the finite element method, and the associated errors can be easily controlled. The authors were thinking about asymptotic error estimators (what happens when the diameter of the largest element goes to zero) that did not account for items 2 and 3. They did not want to bother with the complications caused by numerical integration and the approximation of the domains and boundary conditions, so they declared those features to be crimes. This may have been a clever move but certainly not a helpful one.

Sherlock Holmes investigating variational crimes in Victorian London. Image generated by Microsoft Copilot.

Egregious Variational Crimes

The authors of reference [4] failed to mention the truly egregious variational crimes that are very common in the practice of finite element modeling today and will have to be abandoned if the reliability predictions based on finite element computations are to be established:

1. Using point constraints. Perhaps the most common variational crime is using point constraints for other than rigid body constraints. The finite element solution will converge to a solution that ignores the point constraints if such a solution exists, else it will diverge. However, the rates of convergence or divergence are typically very slow. For the discretizations used in practice, it is hardly noticeable.  So then, why should we worry about it? – Either we are not approximating the solution to the problem we had in mind, or we are “approximating” a problem that has no solution. Finding an approximation to a solution that does not exist makes no sense, yet such occurrences are very common in finite element modeling practice. The apparent credibility of the finite element solution is owed to the near cancellation of two large errors: The conceptual error of using illegal constraints and the numerical error of not using sufficiently fine discretization to make the conceptual error visible.  A detailed explanation is available in reference [5], Section 5.2.8.
2. Using point forces in 2D and 3D elasticity (or more generally in 2D and 3D problems). In linear elasticity, the exact solution does not have finite strain energy when point forces are applied. Hence, any finite element solution “approximates” a problem that does not have a solution in energy space.  Once again, divergence is very slow. When point forces are applied, element-by-element equilibrium is satisfied, and the effects of point forces are local, whereas the effects of point constraints are global. Generally, it is permissible to apply point forces in the region of secondary interest but not in the region of primary interest, where the goal is to compute quantities that depend on the derivatives, such as stresses and strains [5].
3. Using reduced integration. At the time of the publication of their book [4], Strang and Fix could not have known about reduced integration which was introduced a few years later [6]. Reduced integration was justified in typical finite element modeling fashion: Low-order elements exhibit shear locking and Poisson ratio locking. Since the elements that lock “are too stiff,” it is possible to make them softer by using fewer than the necessary integration points. The consequences were that the elements exhibited spurious “zero energy modes,” called “hourglassing,” that had to be controlled by various tuning parameters. For example, in the Abaqus Analysis User’s Manual, C3D8RHT^(S) is identified as an “8-node trilinear displacement and temperature, reduced integration with hourglass control, hybrid with constant pressure” element. Tinkering with the integration rules may be useful in the art of finite element modeling when the goal is to tune stiffness relationships (as, for example, in crash dynamics models), but it is an egregious crime in finite element analysis because it introduces a source of error that cannot be controlled by mesh refinement, or increasing the polynomial degree, and makes a posteriori error estimation impossible.
4. Reporting computed data that do not converge to a finite value. For example, if a domain has one or more sharp reentrant corners in the region of primary interest, then the maximum stress computed from a finite element solution will be a finite number but will tend to infinity when the degrees of freedom are increased. It is not meaningful to report such a computed value: The error is infinitely large.
5. Tricks used when connecting elements based on different formulations. For example, connecting an axisymmetric shell element (3 degrees of freedom per node) with an axisymmetric solid element (2 degrees of freedom) involves tricks of various sorts, most of which are illegal.

Takeaway

The deeply ingrained practice of finite element modeling has its roots in the pre-science period of the development of the finite element method. To meet the current reliability expectations in numerical simulation, it will be necessary to routinely perform solution verification. This is possible only through the science of finite element analysis, respecting the rules of variational calculus. When thinking about digital transformation, digital twins, certification by analysis, and linking simulation with artificial intelligence tools, one must think about the science of finite element analysis and not the art of finite element modeling rooted in pre-1970s thinking.

References

Related Blogs:

• Where Do You Get the Courage to Sign the Blueprint?
• A Memo from the 5th Century BC
• Obstacles to Progress
• Why Finite Element Modeling is Not Numerical Simulation?
• XAI Will Force Clear Thinking About the Nature of Mathematical Models
• The Story of the P-version in a Nutshell
• Why Worry About Singularities?
• Questions About Singularities
• A Low-Hanging Fruit: Smart Engineering Simulation Applications
• The Demarcation Problem in the Engineering Sciences
• Model Development in the Engineering Sciences
• Certification by Analysis (CbA) – Are We There Yet?
• Not All Models Are Wrong
• Digital Twins
• Digital Transformation
• Simulation Governance

By Dr. Barna Szabó
Engineering Software Research and Development, Inc.
St. Louis, Missouri USA

Digital transformation, digital twins, certification by analysis, and AI-assisted simulation projects are generating considerable interest in engineering communities. For these initiatives to succeed, the reliability of numerical simulations must be assured. This can happen only if management understands that simulation governance is an essential prerequisite for success and undertakes to establish and enforce quality control standards for all simulation projects.

The idea of simulation governance is so simple that it is self-evident: Management is responsible for the exercise of command and control over all aspects of numerical simulation. The formulation of technical requirements is not at all simple, however. A notable obstacle is the widespread confusion of the practice of finite element modeling with numerical simulation. This misconception is fueled by marketing hyperbole, falsely suggesting that purchasing a suite of software products is equivalent to outsourcing numerical simulation.  

At present, a very substantial unrealized potential exists in numerical simulation. Simulation technology has matured to the point where management can realistically expect the reliability of predictions based on numerical simulations to match the reliability of observations in physical experimentation. This will require management to upgrade simulation practices through exercising simulation governance.

The Kuhn Cycle

The development of numerical simulation technology falls under the broad category of scientific research programs, which encompass model development projects in the engineering and applied sciences as well. By and large, these programs follow the pattern of the Kuhn Cycle [1] illustrated schematically in Fig. 1 in blue:

A period of pre-science is followed by normal science. In this period, researchers have agreed on an explanatory framework (paradigm) that guides the development of their models and algorithms.  Program (or model) drift sets in when problems are identified for which solutions cannot be found within the confines of the current paradigm. A program crisis occurs when the drift becomes excessive and attempts to remove the limitations are unsuccessful. Program revolution begins when candidates for a new approach are proposed. This eventually leads to the emergence of a new paradigm, which then becomes the explanatory framework for the new normal science.

The Development of Finite Element Analysis

The development of finite element analysis followed a similar pattern. The period of pre-science began in 1956 and lasted until about 1970. In this period, engineers who were familiar with the matrix methods of structural analysis were trying to extend that method to stress analysis. The formulation of the algorithms was based on intuition; testing was based on trial and error, and arguing from the particular to the general (a logical fallacy) was common.   

Normal science began in the early 1970s when the mathematical foundations of finite element analysis were addressed in the applied mathematics community. By that time, the major finite element modeling software products in use today were under development. Those development efforts were largely motivated by the needs of the US space program. The developers adopted a software architecture based on pre-science thinking. I will refer to these products as legacy FE software: For example, NASTRAN, ANSYS, MARC, and Abaqus are all based on the understanding of the finite element method (FEM) that existed before 1970.

Mathematical analysis of the finite element method identified a number of conceptual errors. However, the conceptual framework of mathematical analysis and the language used by mathematicians were foreign to the engineering community, and there was no meaningful interaction between the two communities.

The scientific foundations of finite element analysis were firmly established by 1990, and finite element analysis became a branch of applied mathematics. This means that, for a very large class of problems that includes linear elasticity, the conditions for stability and consistency were established, estimates were obtained for convergence rates, and solution verification procedures were developed, as were elegant algorithms for superconvergent extraction of quantities of interest such as stress intensity factors. I was privileged to have worked closely with Ivo Babuška, an outstanding mathematician who is rightfully credited for many key contributions.

Normal science continues in the mathematical sphere, but it has no influence on the practice of finite element modeling. As indicated in Fig. 1, the practice of finite element modeling is rooted in the pre-science period of finite element analysis, and having bypassed the period of normal science, it had reached the stage of program crisis decades ago.

Evidence of Program Crisis

The knowledge base of the finite element method in the pre-science period was a small fraction of what it is today. The technical differences between finite element modeling and numerical simulation are addressed in one of my earlier blog posts [2]. Here, I note that decision-makers who have to rely on computed information have reasons to be disappointed. For example, the Air Force Chief of Staff,  Gen. Norton Schwartz, was quoted in Defense News, 2012 [3] saying: “There was a view that we had advanced to a stage of aircraft design where we could design an airplane that would be near perfect the first time it flew. I think we actually believed that. And I think we’ve demonstrated in a compelling way that that’s foolishness.”

General Schwartz expected that the reliability of predictions based on numerical simulation would be similar to the reliability of observations in physical tests. This expectation was not unreasonable considering that by that time, legacy FE software tools had been under development for more than 40 years. What the general did not know was that, while the user interfaces greatly improved and impressive graphic representations could be produced, the underlying solution methodology was (and still is) based on pre-1970s thinking.

As a result, efforts to integrate finite element modeling with artificial intelligence and to establish digital twins based on finite element modeling will surely end in failure.

Paradigm Change Is Necessary

Paradigm change is never easy. Max Planck observed: “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” This is often paraphrased, saying: “Progress occurs one funeral at a time.” Planck was referring to the foundational sciences and changing academic minds.  The situation is more challenging in the engineering sciences, where practices and procedures are often deeply embedded in established workflows and changing workflows is typically difficult and expensive.

What Should Management Do?

First and foremost, management should understand that simulation is one of the most abused words in the English language. Furthermore:

• Treat any marketing claim involving simulation with an extra dose of skepticism. Prior to undertaking projects in the areas of digital transformation, certification by analysis, digital twins, and AI-assisted simulation, ensure that the mathematical models produce reliable predictions.
• Recognize the difference between finite element modeling and numerical simulation.
• Understand that mathematical models produce reliable predictions only within their domains of calibration.
• Treat model form and numerical approximation errors separately and require error control in the formulation and application of mathematical models.
• Do not accept computed data without error metrics.
• Understand that model development projects are open-ended.
• Establish conditions favorable for the evolutionary development of mathematical models.
• Become familiar with the concepts and terminology in reference [4]. For additional information on simulation governance, I recommend ESRD’s website.

References

Related Blogs:

• Where Do You Get the Courage to Sign the Blueprint?
• A Memo from the 5th Century BC
• Obstacles to Progress
• Why Finite Element Modeling is Not Numerical Simulation?
• XAI Will Force Clear Thinking About the Nature of Mathematical Models
• The Story of the P-version in a Nutshell
• Why Worry About Singularities?
• Questions About Singularities
• A Low-Hanging Fruit: Smart Engineering Simulation Applications
• The Demarcation Problem in the Engineering Sciences
• Model Development in the Engineering Sciences
• Certification by Analysis (CbA) – Are We There Yet?
• Not All Models Are Wrong
• Digital Twins
• Digital Transformation

By Dr. Barna Szabó
Engineering Software Research and Development, Inc.
St. Louis, Missouri USA

Digital transformation is a multifaceted concept with plenty of room for interpretation. Its common theme emphasizes the proactive adoption of digital technologies to reshape business practices with the goal of gaining a competitive edge. The scope, timeline, and resource allocation of digital transformation projects depend on the specific goals and objectives. Here, I address digital transformation in the engineering sciences, focusing on numerical simulation.

Digital Technologies in the Engineering Sciences

Digital technologies have been integrated into the engineering sciences since the 1950s.  The adoption process has not been uniform across all disciplines. Some fields (like aerospace) adopted technologies early, while others were slower to change. The development and adoption of these technologies are ongoing. Engineering today is increasingly digital, and innovations are constantly changing the way engineers approach their work. Here are some important milestones:

Early Adoption (1950s-1970s)

• Mainframe computers were used for engineering calculations that would have been impossible or extremely time-consuming to perform by hand.
• Numerical control (NC) machines used punched tape or cards to control tool movements, streamlining machining processes.
• Early Computer-Aided Design (CAD) systems revolutionized drafting in the 1960s. They allowed engineers to create and manipulate drawings on a computer, making design iterations much faster than previously possible.

Period of Rapid Growth (1980s-1990s)

• Affordable Personal Computers (PCs) made computing power accessible to individual engineers and small firms.
• Development of CAD software brought 3D modeling from specialized applications into mainstream design.
• Finite Element Modeling software became commercially available, allowing engineers to perform structural and strength calculations.
• The mathematical foundations of the finite element method (FEM) were established, and finite element analysis (FEA) became a branch of Applied Mathematics.

Post-Millennial Development  (2000s-Present)

• Cloud-based solutions offer scalable computing power and collaboration tools, making complex calculations accessible without massive hardware investment.
• Building Information Modeling (BIM) revolutionized the architecture, engineering, and construction (AEC) industries.
• Internet of Things (IoT): Networked sensors and devices provide engineers with real-time data to monitor structures, predict maintenance needs, and optimize operations.
• Additive Manufacturing (3D Printing) allows for the rapid creation of complex prototypes and even functional end-use parts.

Given that digital technologies have been successfully integrated into engineering practice, it may appear that not much else needs to be done. However, important challenges remain, and there are many opportunities for improvement. This is discussed next.

Outlook: Opportunities and Challenges

Bearing in mind that the primary goal of digital transformation is to enhance competitiveness, in the field of numerical simulation, this translates to improving the predictive performance of mathematical models. Ideally, we aim to reach a reliability level in model predictions comparable to that of physical experimentation. From the technological point of view, this goal is achievable: We have the theoretical understanding of how to maximize the predictive performance of mathematical models through the application of verification, validation, and uncertainty quantification procedures. Furthermore, advancements in explainable artificial intelligence (XAI) technology can be utilized to optimize the management of numerical simulation projects so as to maximize their reliability and effectiveness.  

The primary challenge in the field of engineering sciences is that further progress in digital transformation will require fundamental changes in how numerical simulation is currently understood by the engineering community and how it is practiced in industrial settings. It is essential to keep in mind the differences between finite element modeling and numerical simulation. I explained the reasons for this in an earlier blog post [1]. The art of finite element modeling will have to be replaced by the science of finite element analysis, and the verification, validation, and uncertainty quantification (VVUQ) procedures will have to be applied [2].

Paradoxically, the successful early integration of finite element modeling practices and software tools into engineering workflows now impedes attempts to utilize technological advances that occurred after the 1970s. The software architecture of legacy finite element codes was substantially set by 1970, based on understanding the finite element method that existed at that time. Limitations of the software architecture prevented subsequent advances, such as a posteriori error estimation in terms of the quantities of interest and control of model form errors, both of which are essential for meeting the reliability requirements in numerical simulation. Abandoning finite element modeling practices and embracing the methodology of numerical simulation technology is a major challenge for the engineering community.

The “I Believe” Button

An ANSYS blog [3] tells the story of a presentation made to an A&D executive. The presentation was to make a case for transforming his department using digital engineering. At the end of the presentation, the executive pointed to a coaster on his desk. “See this? That’s the ‘I believe’ button. I can’t hit it. I just can’t hit it. Help me hit it.” Clearly, the executive was asking for convincing evidence that the computed information was sufficiently reliable to support decision-making in his department. Put in another way, he did not have the courage to sign the blueprint on the basis of data generated by digital engineering. What it takes to gather such courage was addressed in one of my earlier blogs [4]. Reliability considerations significantly influence the implementation of simulation process data management (SPDM).

Change Is Necessary

The frequently cited remark by W. Edwards Deming: “Change is not obligatory, but neither is survival,” reminds us of the criticality of embracing change.

References

Related Blogs:

• Where Do You Get the Courage to Sign the Blueprint?
• A Memo from the 5th Century BC
• Obstacles to Progress
• Why Finite Element Modeling is Not Numerical Simulation?
• XAI Will Force Clear Thinking About the Nature of Mathematical Models
• The Story of the P-version in a Nutshell
• Why Worry About Singularities?
• Questions About Singularities
• A Low-Hanging Fruit: Smart Engineering Simulation Applications
• The Demarcation Problem in the Engineering Sciences
• Model Development in the Engineering Sciences
• Certification by Analysis (CbA) – Are We There Yet?
• Not All Models Are Wrong
• Digital Twins