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A nonlinearity in the past

Zhigang Suo's picture

I'm in Washington DC attending a small workshop entitled Understanding and Exploiting Nonlinearity.  Yesterday several talks described recent developments of nonlinear dynamics, the kind of phenomena that can be described by a set of nonlinear ordinary differential equations.  Years ago, when the theory of chaos was in vogue, I looked at several textbooks on nonlinear dynamics, and tried to apply a few elementary ideas to evolving structures in materials.  This time I learned that many of the esoteric ideas of nonlinear dynamics have found applications in modeling natural phenomena and creating new devices.  Perhaps it is a good time to relearn nonlinear dynamics.

Talks today are on nonlinear phenomena in various fields.  I'll be talking about nonlinear field theory of active soft materials.  The speaker now is talking about semiconductor lasers.  My mind starts to wander...

When I came to the United States in 1986 as a graduate student, the International Office of the university matched me with a host family, Ed and Jane.  Ed was a professor at MIT, and Jane was a housewife.  They invited me several times to dinner.  They took me to MIT campus and museums in Boston.  When we met, Jane did most talking, and Ed remained quiet.  For my first Thanksgiving in the United States, they invited me for dinner at their apartment on Memorial Drive.  Their grown children also came.  After dinner we went for a walk along the Charles River.  I had a conversation with his son.  I mentioned that I was an engineering student and liked mathematics.  The son told me that his father, Ed, was a meteorologist and also liked mathematics.

The significance of the conversation did not register in my mind until years later when I began to be interested in nonlinear dynamics.  I finally made the connection:  Ed was Edward Lorenz, father of chaos theory and butterfly effect.


Mike Ciavarella's picture


thanks for your post.

Of course, the case of Lorenz is interesting, where computers were needed already then to solve non-linear equation, and the fact that they were weak, induced Lorenz to find the chaos theory (his punching card input did not contain the same digit as the output in the machine, so every time a valve broke in his computer, he started from a different condition). He was surprised at first that the results would be so bizarre, then he realized some cases are very initial condition sensitive.

In Solid Mechanics, we have of course used non-linear problems for a long time, but MAINLY we have tried to reduce the non-linearity to a manageable form. There are various forms of non-linearity.  In plasticity, we use very often the J2 theory, to remove some difficulties due to path-dependence, and that led to J-integral, HRR field, etc. In construction engineering, I was amazed to talk the other day to a ARUP engineer --- they do everything by hand!

I am surprised to be honest on how blindly we teach some of our students to use computer programs to do non-linear problems. Commercial FEM software companies have flourished, and they give the impression that one could do "virtual prototyping" even of the most complex non-linear problem, including plasticity, viscous effect, dynamic effects, large deformation, even atomistic simulations of a car crash !!!!   Do you beleive it or is it a videogame?

What I mean is, given computers are weak even today, the equivalent of Lorenz discover could be that some computer predicts something completely unrealistic to us, then we find some hints for new theories to be badly needed. Of course, not many are lucky as Lorenz.

A second observation is that we learn more by failure, than by success.  You know I have recently been frustrated by how publishing works, but NEGATIVE results should be published.  I know of some good US author who has found, for example, that rate-state dependent friction can be obtained easily with tuning DIFFERENT parameters. That results doesn't look good for a paper, but it would be CRUCIAL to develop a new theory.  He doesn't want to publish it.  His name is AB.

Regards, Mike


michele ciavarella


Thanks for sharing. I'll admit that I do not really knw much about the mechanics you were mentioning, but your story about Dr Lorenz put a big smile on my face. 



Thanks for sharing, too. Your remarks about engineering software especially struck me as interesting. There is certainly an epidemic of blind trust in computer programs' outputs, but I must say that my confidence in computer simulations has only gone up over time. (As a caveat, perhaps I am one of those impressionable students Wink .) It is truly astonishing the agreement between full-scale tests and computer simulations by competent modelers. 

I am a very skeptical person and finite element simulations are not immune to my skepticism. It is a lot to swallow that we could be so good at simulating the universe and I have thought that it was a whole lot of voodoo at times. However, I've seen how amazingly well experiments and models can match up, even for highly nonlinear, highly dynamic, very complicated problems and am now in large part a believer. 

That isn't to say there isn't a lot of trash. As the old adage about computers goes—Garbage In, Garbage Out. However, it seems to me from my experiences that even for very complex problems, it is possible to put good inputs and get good outputs. 

I don't exactly know what my point is in saying this. I suppose I just mean to indicate that some of us do have a lot of confidence in the methods we use in simulating these problems, but it isn't blind belief from being taught, but rather we overcame our skepticism by seeing the methods work.



Zhigang Suo's picture

Dear Mike and Mike:  I agree with your sentiments.  Many of us are ambivalent about computer simulations, just as we are ambivalent about mathematics and all those long equations. Even farther back, people were skeptical about the use of written language to communicate the Word of God.  Compared to spoken language, written language was thought to be static and susceptible to misinterpretation.

Nonetheless we just have this curiosity to learn about things around us, and this urge to tell others what we have learned, in whatever methods possible, however imperfect the methods are.  By using all modes of communicating and learning, we will exploit their strength and expose their weakness.  Nature does not exist for us to comprehend. We'll just use whatever tool that aids us, be it the microscope or the computer.

The availability of inexpensive computer simulation has provided a new opportunity to theoreticians:  we can now create theories for more complicated phenomena, knowing that the computer will do the thankless work:  computing.  In lectures, we no longer need to apologize for showing the audience long equations.  we just remind the audience that these equations are really intended for the computer, who does not have emotion, has no use for our intuition, and has no fear for equations.  A purpose of these equations is to help the computer help us understand the world.  And a purpose of the lectures is to tell people how we help the computer.

For anyone who has ever heard the words "finite element",

A key reason why people are skeptical about finite element software is that they don't understand the logic behind it.  In the case of simulation, not knowing how something works can mean not knowing if something works.  This can be dangerous, because designing structures with such software might result in loss of life on a large scale.

This is precisely why

1. I remain skeptical about finite element analysis, in spite of what I was taught in grad school.
2. I left my job at a prestigious structural engineering firm, in spite of having nothing to fall back on.
3. I spent the past two years searching for answers out of pocket, in spite of my $23K in student loans.
4. I am reluctant to believe what anyone tells me, even those with a Ph.D., a high GPA, and an ability to reproduce everything that is currently supported by the scientific establishment.
5. I am unable to work for people who tell me what to do without proof that it should be done that way.

I and other "people of unusual creativity and independence"* can find answers and provide an understanding of how things work, but only if we are given the time to seek those answers out, the freedom from the distractions imposed upon us, and the resources we need to stay alive.

David M. Cooper

*Lee Smolin, "Why No 'New Einstein'?"

Mike Ciavarella's picture

For Mike

I know some simulations "look realistic", and hence convincing. But remember that ALSO videogames sound more and more realistic.  It is a lot about "tuning" parameters, which sometimes are "magic" and unique to a given simulation.  So what on the next simulation?

I suspect we should divide , in analogy with chaotic and non chaotic being different for the sensitivity to initial conditions, in virtual prototyping we should differentiate between "sensitive to tuning parameters" and non-sensitive.

I have seen different codes in Fracture Mechanics even produced by remarkably well known groups, like NASA, ONERA, etc., turn results COMPLETELY different from the SAME nominal conditions.  Read the figure in my presentation

A seminar at Paris VI inst. d'alembert - One, no one, and one hundred thousand crack propagation equations: thursday June 5th. 1

I think these days there are software which COMPARE between results of different codes, and see the ROBUSTNESS of the response.  While some naif student beleive Commercial softwares, the Top management is well aware of the problem.

I think there are at least 2 journals for the industry dedicated to this aspect of BENCHMARKING


both are part of NAFEMS.

Welcome to Nafems

Welcome to the NAFEMS website, dedicated to providing independent information on engineering analysis and simulation.

michele ciavarella

For sure, there are some simulations that are not adequate for modeling the underlying physical phenomena. There is a lot of magic and video games in the simulation industry. The problem of finding useless solutions is certainly not unique to the computer simulation technique, but it can be very tempting to believe a simulation that cannot really solve the problem the engineer is tasked to solve. Sometimes the physics is not modeled by the tools at hand; sometimes the modeler does not or cannot provide enough information to find the more valuable solution when the solution is not especially stable.

However, what I was saying was that FEM software is equipped to handle a wide range of engineering problems. It is not some miracle or magic trick that the method works—we (usually) have derived why a method can solve a problem. These methods are usually implemented rather transparently in commercial codes.

(Indeed, a greater level of transparency would be nice. I am a big fan of open source software: I'm posting my comment Creative Common licensed on an open source CMS using an open source web browser running on an open source operating system. I believe transparency solves many problems created by secrecy in this realm.)

Thanks for all your comments on this issue. I do share your skepticism, but have not reached quite the same level of cynicism about these simulations. I do not believe I've been fooled by the high-quality simulations I've seen. I am a very critcally-minded person and always keep in mind the ways simulations' qualities could be misrepresented when would-be impressive results are shown to me or found by me.



Mike Ciavarella's picture

The trouble start when we have NO underlying equations.  Friction, crack propagation, behaviour of material, especially under high strain, impact.  These are all unknown.  Let alone turbulence.  How you can fit to a model?

Only tuning it to be reasonable.   There are commercial softwares which are SOLD to industry in the HOPE they address these difficult tasks.  But is like selling old drugs in the Far West.  People buy it, and expensively, but they are then disappointed to find that these things are no better than hand calculation.

This is why ARUP designs even skycrapers by hand!  Really good industry beleives still in single geniuses, than in hundreds of FEM users....

michele ciavarella

Yes, there are some phenomena where we do not really have good engineering equations to use. Friction isn't really one of them—we have really good equations for some types of friction. "Behaviour of material" is so general I do not know what to say about it. There is some theoretical background for impact simulations. Nonetheless, the parts of the finite element method which we arrive at with no fundamentals are of course not trustworthy.

Even the best applications of the FEM, like all engineering techniques, are subject to being used right. Sometimes they are no better than a hand calculation, but sometimes they are much, much better. Often, of course, closed form solutions are not feasible and approximations of theoretical results are necessary, by hand with tables or charts or guesses or by the FEM.

It's a matter of using the right tool for the right job. If I have a very deep beam, I would be wrong to use Bernoulli beam theory. If I had a problem a computer tool is not equipped to solve, I would be wrong to use such a tool.

As a point of fact, are you completely sure that Arup does not use computer tools for design? I've talked to structural engineers from Arup and I believe they described some of the computer tools they used. Their website confirms they use and distribute FEM software for complex simulations, such as impact.

The FEM is an invaluable tool for structural design. Linear, static FE analyses are standard for structural design because they allow designers to analyse statically indeterminate structures very simply. The results of such analyses are easily verified and are extremely reliable when the method is used right. Structures are more easily designed with redundancy and less menial analysis work on the part of engineers.  I find it hard to believe Arup would not take advantage of such technologies.

Mike Ciavarella's picture

Zhigang and Mike, see this for example: 

FENet Survey Into Barriers To The Effective Use Of Finite Element
Analysis in Industry

Dr. Jim Wood, University of Strathclyde, UK

The FENET world-wide survey, which sought to establish the
significance of barriers to the effective use of finite element
analysis across a range of industry sectors, including Aerospace,
Land Transport, Biomedical, Civil Construction, Consumer Goods,
Marine and Offshore, Power and Pressure Systems and Process &
Manufacture, ran from November 2002 - July 2003.

The survey, consisting of over 450 questions, included a wide range
of issues covering education and awareness, the staffing of FE
projects, the cost of FE products; support-related matters in the
widest sense, and an extensive list of functionality-related
matters, including integration of the analysis function. Particular
emphasis was placed on the topics of multi-physics & analysis
technology, life extension & durability and product &
system optimization, as these are identified themes within the
FENet project.

In addition to documenting current state of practice, it is
anticipated that the results will provide the basis for other
useful business benefits including:

  • making a useful input to technical strategies and roadmaps;

  • providing a focus for identifying potential areas of competitive

  • facilitating the identification of areas for possible code

  • assisting with the identification of differences in awareness and
    use of technology across the industry sectors examined.

The results from the extensive questionnaire have already informed
the future plans of the various FENet Industry Coordinators and
some of the more interesting results have been reproduced in the
following discussion.

Background to the Survey

One of the main objectives of the FENet project is to improve both
the quality of industrial applications of the technology and the
level of confidence that can be placed in results. It was perhaps
not surprising therefore that initial technical discussion should
focus on the barriers to achieving this across the various industry
sectors represented in the project. The Education and Dissemination
Technical Workshops held in Copenhagen and Zurich considered the
general issues and were instrumental in the formulation of the

General Information

Over 1.300 replies were received from more than 40 countries,
although most responses came from the UK and US. Around 50 % of
respondents were industrial users, 20 % were consultants, 15 %
academics, with the remainder being made up of software vendors and
researchers. Of these industrial users, there were 20 % each from
Aerospace, Land Transport and Process & Manufacturing; 10 %
each from Power & Pressure Systems, Civil Engineering &
Construction and Marine & Offshore, with the remainder from
Consumer Goods and BioEngineering.

From the replies, it was clear that the survey had mainly reached
experienced users, with Linear Statics being the most common type
of analysis being carried out, closely followed by Non-linear
Statics. Interestingly, 40 - 50 % of respondents (range indicates
difference between EU and North American responses) worked in
“business units” with less than 5 users and over 50 %
of respondents had Unix platforms. Over 60 % report that their use
of FEA is growing and 75 - 80 % did not see an end to the ever
increasing level of detail and complexity in models.

User Environment

In terms of the user environment, around 40 % of respondents work
in business units that operate internal user group meetings; around
60 % employ some form of mentoring and around 45 % disseminate best
practice (precisely how is not known). Altogether, around 95 % of
respondents made use of personal contacts, thus emphasizing the
importance of networking. Quite surprisingly perhaps, 70 - 80 % of
respondents indicated that they use newsgroups and discussions
forums. Interestingly, over 80 % felt that some way of capturing
and re-using experience would be useful. Perhaps there is a
research opportunity to re-visit the area of knowledge management
and expert systems, that failed to deliver some 10 – 20 years

Systems and Supply

Positively, almost 90 % of respondents felt that their FE
investment had been effective ... but then again the survey
didn’t really get to the non-users of the technology (and
possible former users). Vendors received a further pat on the back,
with almost 90 % of responses for vendor support ranging from
important to vital.

Only 10 - 20 % felt that this support was inadequate and more than
20 - 25 % felt that it was excellent. Consultants did not fare as
well and while around 50 % put some degree of importance on support
from consultants, only 10 % rated the support as excellent with 20
% rating it as inadequate.

In terms of non-purchase of an FE system (users of the technology
only were surveyed remember), 75 % of respondents indicated that
cost was a common or major issue, 55 % reported nonrelevance of the
technology as a common or major issue and 65 % reported unclear of
the cost benefits as a common or major issue. Additionally, 44 %
blamed current trading conditions as a common or major issue, 42 %
indicated that the level of complexity was a common or major issue
for the products they had considered and 40 % highlighted staffing
issues as a common or major issue.

In terms of non-renewal of FE system maintenance contracts, 53 % of
respondents indicated that a common or major issue was the fact
that cost outweighed the business benefits, while 28 % reported
perceived inadequate support as a common or major issue. The
availability of in-house expertise obviously has some bearing on
the need to use hotline support and 38 % reported the availability
of adequate inhouse expertise as a common or major issue for
non-renewal. On the other hand, 26 % highlighted loss of in-house
staff expertise as a common or major issue for nonrenewal. The fact
that software upgrades were not required was a common or major
issue with 26% of respondents.

With respect to getting the most out of the technology, 60 %
reported infrequent use as a common or major issue and similarly
over 70 % felt that ease of use was either very important or vital.
Clearly there is still a demand for easier to use software.

New functionality in the finite element systems results in little
benefit if they are difficult to use and/or staff do not have the
time to explore the functionality and around 65 % of respondents
reported that time pressure was a common or major issue in terms on
not getting the most out of the technology. This latter issue also
relates to staff development, which is addressed below. Also
related, is the fact that over 70 % of respondents felt that the
lack of understanding of the business benefits of FEA by
management, resulted in some form of barrier.

Staffing and Development

In terms of staffing, 45 - 40 % felt that the recruitment of
suitably qualified staff was a significant or very significant
barrier to their use of the technology and around 45 % blamed staff
turnover as a common or major issue. Despite this, 50 - 45 %
reported that their organisations don’t subcontract work and
60 - 65 % don’t employ on-site contractors.

The responses to staff development questions would seem to indicate
that there is perhaps a market for education and training resources
aimed at self-learning, as 80 % of respondents felt that time off
the job while training was some form of barrier. This point was
reinforced by the fact that a similar percentage saw the lack of
adequate and convenient training as some form of barrier.

In addition, 85 - 80 % of respondents felt that the cost of
training was a barrier and 30 - 25 % felt that the poor quality of
training was either a significant or very significant barrier.
Similarly, around 65 % blamed lack of investment in training as a
common or major issue in not using their system effectively. The
fact that 30 - 25 % of respondents felt that the education of new
graduates was either a significant or very significant barrier to
the effective use of FEA may be of concern to those involved in


Further trends in academia also give cause for concern and indicate
a sector under change. It is perhaps an indication of cause and
effect that 65 % of academic respondents reported increased
competition amongst institutions, whilst 61 % reported increased
collaboration. Of greater concern to the well-being of engineering
and the future supply of wellqualified graduates to industry, is
that around 48 % reported merging amongst science and engineering
departments, 28 % reported closure and 49 % reported a loss in
facilities. While 60 % reported a loss of staff in science and
engineering, only 40 % reported a loss in the number of specialist
subjects in degrees.

This contraction is perhaps in direct response to the fact that 58
% reported a reduction in the number of students entering
undergraduate engineering and related courses, with the same figure
for postgraduate courses.

Approaching 70 % of the academic respondents reported a reduced
mathematical ability in school leavers and 36 % reported a
reduction in FE and simulation in degrees.


The number of detailed questions relating to the general topic of
integration of the analysis and simulation function into the wider
business enterprise is testimony enough to the importance placed on
this area by the various industry coordinators involved in the
development of the survey.

The various replies confirm the importance. Around 50 % of
respondents rated simplification and defeaturing as very important
to vital, while over 40 % rated reuse of data and results as very
important to vital. In addition, 50 % rated FE to CAD updating as
very important to vital and over 40 % rated re-use of data and
results as very important to vital.

Although only 50 % of respondents overall, rated use & control
of legacy data as very important, this percentage was much higher
in some business sectors.

Technology and Industry

The technology specific questions in the survey were structured
around the notion of Technology Readiness Levels (TRL’s),
which use ratings from 0 (low) -9 (high) to indicate both the
Priority and the Maturity of the particular technology in the
respondent’s business sector. The highest Priority for
Durability and Life Extension @ 6.2 was fatigue life prediction and
assessment, whilst the lowest Maturity @ 3.3 was
damage/deterioration modelling and assessment. The highest Priority
for Product and System Optimisation (PSO) @ 5.5 was application of
structural and system optimisation tools, whilst the lowest
Maturity @ 2.9 was use of decision support tools for management
issues. In addition, 65 % of PSO respondents felt that user
education and training was a barrier to the effective use of the
technology and 55 % of PSO respondents also felt that management
education and awareness was lacking.

The highest Priority for Multi- Physics and Analysis Technology @
7.1 was automatic meshing, whilst the lowest Maturity @ 1.74 was
magnetic hydrodynamics. This small sample of results from the three
FENet RTD areas were drawn from a questionnaire report based on all
responses. Reports based on responses from specific industry
sectors were supplied to the various FENet Industry Coordinators
and these reports in turn are reflected in the FENet Industry
Requirement Reports. In many cases the “overall” and
“industry-specific” conclusions were similar, but in
some cases, marked differences were apparent.

 In the Aero Sector report, Failure Criteria for Advanced
Materials had a Priority of 7 and a Maturity of 4. Probabalistic
Methods was also highlighted as being important. In the Land
Transport Sector, Modelling of Connections also had a Priority of 7
and a Maturity of 4. In the Civil & Construction Sector,
Material Models for Buildings had a Priority of 7 and a Maturity of
4.5. In the Power & Pressure Systems Sector, Design by Analysis
appeared as a significant issue and in the Process &
Manufacturing Sector, Material Data figures prominently.


The general issue of validation is clearly important to analysis
and simulation. It is rather unfortunate therefore that this
section of the questionnaire was probably less than satisfactory in
terms of how the questions were posed and how users were
constrained in their responses. Nevertheless, it is suggested that
95 % of respondents felt that poor material data for input and
assessment was important to vital as a validation issue and that 95
% also saw the general lack of correlation with test as an
important to vital issue.

Unfortunately the questionnaire did not allow users to select more
than one method of results validation used. As a result, the
conclusions in this area should probably be interpreted as the
users most popular method of model validation. It will be
reassuring to many that experimentation and test is alive and well,
with almost 40 % of respondents indicating that they use physical

Surprisingly, but perhaps evidence of the complexity of analyses
being carried out, only just over 30 % indicated hand calculations.
Also perhaps surprising is that less than 4 % indicated other FE
codes and around 2 % indicated nothing at all. The latter result
perhaps being a protest at the length of the questionnaire!


michele ciavarella

ericmock's picture



I would be interested to hear more about the workshop and what kinds of things were presented and who presented them.  Is there information online?


When I was just starting my career as a professor I realized that I needed something to summarize my research interests.  Trying to figure this out, I realized that my interests were in "Understanding and Exploiting (Structural) Nonlinearity."  Ever since, this is what I tell my peers I do.  It's really the only thread that seems to tie together the vary disjoint research topics I've explored.


Having started my research career (for an MS thesis) doing nonlinear vibrations/dynamics I turned my attention to solid mechanics after moving to UCB for my PhD and taking continuum mechanics from Paul Naghdi.  However, I continued to work in a lab (Dan Mote's Dynamic Stability Lab) that focused mostly on dynamics and vibration.  I have thus tried to use my background in these areas to do unique work.  This however has often left me feeling in between two communities and not really a part of either.


Zhigang's original post reminds me of an interested discussion Joe Cusumano and I had during the time we were team teaching a rigid body mechanics graduate course [1].  We were discussing some interesting results I was getting for the deformation of rods in self-generated force fields.  This lead to a discussion of the kinetic analog between elastica equations and pendulum oscillation, and finally to a discussion of using force fields to introduce spatial chaos in a rod.  I'm not sure why you'd want to, but....




[1]  This was a great experience as Joe is more of a nonlinear dynamics person and I'm more of a solid mechanics person.  We had lots of educational philosophical debates throughout the semester about many topics at the intersection of nonlinear dynamics and solid mechanics.  Team teaching was a great experience that I would recommend although certainly more than half the work.

Zhigang Suo's picture

Dear Eric:  Thank you very much for relating your own nonlinear experience. The organizers of the workshop asked the participants not to distribute the content of the workshop.  There is no online documentation of the workshop.

ericmock's picture

Zhigang, I hate when organizers take this attitude (unless there are national security concerns).  It really is detrimental to building a research community and leads some people to feel that there is an 'insiders mafia' in which they have been excluded.  Is there anything at all you can say (publicly or privately)?

Prof. Suo, A great topic! In my opinion, nonlinearity is magic and mysterious. When I studied in Chinese Academy of Science, I attended a half-year course titled "computer simulations of modern physics problems". One homework is to solve the group of differential equations of Edward Lorenz by programming. The teacher told me my results are right, but I could not understand why at all. Even though many discussions with the teacher who is physicist, I still failed to reach an understanding.

This is my terrible nonlinear experience! I am waiting for wonderful stories about nonlinearity and especially their explanations, mechanical but not physical.

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