Extended Finite Element Method

Since its introduction in the summer of 1999, the eXtended Finite Element Method (X-FEM) has enjoyed a considerable level of success and popularity from researchers in the computational and applied mechanics communities. In just under six years, several hundred peer-reviewed journal articles dealing with fundamentals or applications of the X-FEM and related methods have appeared in the literature. This post provides some background on the method and comments on its capabilities and usefulness for the broad applied mechanics community. We also welcome questions from the community in the comments area following the post.

The X-FEM grew out of research into meshfree methods by the computational mechanics group at Northwestern University directed by Ted Belytschko. Related methods include the Generalized Finite Element Method (G-FEM) developed at the University of Texas at Austin and Texas A&M University. The basic philosophy of the X-FEM is that features of interest in a problem, for example crack surfaces, phase boundaries, and fluid-structure interfaces, can be represented independently of the finite element mesh. As a result, simulating the evolution of these features is greatly facilitated. This is particularly true when they exhibit changes in topology, such as when multiple crack fronts merge or a single crack front branches. The finite element mesh need not explicitly "fit" these features with the X-FEM, circumventing the need to re-mesh in many cases and facilitating adaptivity in others. The Figure below shows an example of the evolution of a phase boundary in a hydrogel as simulated using a fixed mesh with the X-FEM. The sharp gradients in the deformation (observable in the deformed mesh on the right) have been captured using enrichment.


The basic ideas behind the method are easy to understand. Most finite element approximations to bulk fields (e.g. displacement, temperature) can be expressed as a linear combination of nodal shape functions. These shape functions are only able to represent discontinuities in the bulk fields if the mesh is constructed in a particular way. For example, the classical approach to representing the jump in displacement field across a crack front is to explicitly mesh both crack faces. With the X-FEM, the classical mesh need only overlap the geometry of the crack front and does not need to be carefully aligned with it. The linear combination is then augmented with enrichment functions that capture the jump in displacement field across the crack. Crack growth can in turn be simulated through the identification of additional enriched nodes and a new construction for the enrichment function, a process that is typically much simpler than re-meshing. If additional information about the solution is known---such as the asymptotic behavior of the crack-tip fields---it can also be included in the enrichment to garner coarse-mesh accuracy. The mathematical underpinnings behind this construction---the partition-of-unity method---were established by Ivo Babuska and colleagues at Maryland in the early nineties.

Although the X-FEM was originally designed for linear elastic fracture mechanics, it has since been adapted to a wide range of applications. These include the representation of complex microstructures, multi-phase flow, virtual surgery, and general fluid-structure interaction problems. Recently, many researchers have coupled the X-FEM to the Level-Set Method, a technique for representing surfaces through implicit functions. The combination is a powerful one for simulating evolving boundary value problems. Although similar fixed-grid techniques have been used by the finite-difference community for some time, the variational framework naturally employed by the X-FEM makes the incorporation of enrichment straightforward.

In addition to new applications, research into fundamental issues continues at Northwestern University, Ecole Centrale Nantes, the University of California Davis, and Duke University, among others. Many X-FEM researchers maintain websites dedicated to the method and its applications, and interested readers are encouraged to seek them out.

Nonlinear Finite Elements for Continua and Structures, by Ted Belytschko, Wing Kam Liu, Brian Moran

This is a graduate-level textbook written by established researchers in the field. At amazon.com, the book has receieved 4 stars, and 10 customer reviews.

Current Challenges in Mechanics of Materials

The Second Meeting of the Thin-Air Philosophical Society (TAPS), organized by Professor Demitris Kouris and sponsored by NSF, was held at the University of Wyoming in August of 2005. The presentations were posted on the website of the meeting.

The last morning of the meeting was devoted to a discussion of current challenges in the mechanics of materials. Topics include
• Applications of mechanics and materials
• Tools of the trade
• Mechanics of integrated small structures
• Genes, molecular architectures, and mechanical behaviors
• Integrating mechanics and chemistry
• Mechanics in the field of energy
• Sensing the world
• Multiscale simultaneous design
• Coupling of quantum mechanics with traditional mechanics at small scales; emergence of new computational tools and concepts
• Hierarchical top-down vs. Bottom-up engineering of models for complex systems: beyond multiscale modeling
• Continuum mechanics and thermodynamics

The participants did not aim for an exhaustive list of topics. Rather, examples were discussed, with sufficient diversity to reflect issues of general interest. Individual participants were then assigned to write paragraphs to describe these examples.

Applied Mechanics News: Kyung-Suk Kim won Ho-Am Prize of $200,000

Applied Mechanics News: Kyung-Suk Kim won Ho-Am Prize of $200,000

Professor Carl T. Herakovich won the 2005 Applied Mechanics Award

At the Annual Dinner of the Applied Mechanics Division last November, in Orlando, Florida, Professor Carl T. Herakovich was presented the 2005 Applied Mechanics Award, in recognition of his distinguished contributions to mechanics of fibrous composite materials, and his distinguished service to the mechanics and engineering science community. The text of his acceptance speech follows.

Acceptance Speech
The 2005 Applied Mechanics Division Award
Carl T. Herakovich

Thanks Wing, it is indeed a great honor and pleasure to be recognized by the Applied Mechanics community.

I hold the mechanics community in the highest regards and with the utmost respect. I am always so impressed by the intelligence of the people in this community, their honesty and their candor.

And I can really enjoy being around mechanicians in a social setting. Give them a little wine at dinner and it can be quite a party. I really do enjoy the people in this community. I feel very much at home. (Comment briefly on the dinner in Warsaw at the International Congress in August 2004, and the dinner in DC in Sept. 2005.)

As it happens, you have given me a very nice 50th Anniversary gift. It was 50 years ago, September 1955, that I entered Rose Polytechnic Institute in Terre Haute, Indiana (now Rose-Hulman Institute of Technology) to begin my studies in engineering.

How did I end up spending 50 years in mechanics? Several people had a major impact on my decisions along the way. Professor Richard H. F. Pao at Rose was undoubtedly the first person that peaked my interest in the field. The first class I had from Prof. Pao was in fluid mechanics and my oldest memory of him is the time that I fell asleep in his 8 o’clock fluids class one wintry day while sitting next to a hot radiator with my heavy coat still on. Pao woke me up and left no question that I had embarrassed him by falling asleep in his class. That had a major impression on me, as I never like it when students fell asleep in my class in later years. The class that really made me realize that there was this field called mechanics was an elective that I took from Prof. Pao in the second semester of my senior year. The course was on Advanced Mechanics of Materials out of the old Seely and Smith book.

The next person who influenced my studies in mechanics is clearly my wife Marlene. We met in Terre Haute in Sept. 1957, and married not quite three years later in April 1960. In August 1960, we were preparing to go to Colorado where we both were to have jobs. I then saw an announcement that assistantship were available in Mechanics at The University of Kansas. Now, even though we had only been married about three months and Marlene was about 2.5 months pregnant, she agreed to my wild, out-of-the-blue suggestion that we forego the jobs and go to Kansas so I could study mechanics in graduate school. There had been no previous indication that I would ever want to pursue graduate education. (This change in attitude was undoubtedly influenced by my job at the time working for the Indiana Highway Department.)

After two years in Kansas and two more back at Rose teaching and coaching, I again decided that I would like to go back to graduate school to study more mechanics. Again, Marlene agreed even though we now had two sons. Thus, we ended up in at IIT Chicago (very near our homes in Northern Indiana). There I met Prof. Phil Hodge who became my PhD advisor and mentor. He is a great role model and one who has been very active in AMD, serving on the Executive Committee and as Editor of JAM. He was also active in ASME boards and committees as well as the U. S. National Committee on Theoretical and Applied Mechanics. I have followed in his footsteps in many of these activities.

I have had the good fortune to know and interact with a number of the people who have made a significant impact on mechanics in general and this Division in particular. In addition to Hodge at IIT, fellow students were Ted Belytschko and Bill Saric, both former members of the AMD EC.

During my tenure on the Executive Committee, I was privileged to work with David Bogy, Ben Freund, John Hutchinson, Tom Cruse, Stan Berger, Lallit Anand, Alan Needleman and Tom Hughes. This was a very active time for the EC as we initiated two new ASME medals, the Koiter Medal and the Drucker medal. I had the privilege of informing both Professor Koiter and Professor Drucker that a medal was established in their honor and that they were to be the first recipients.

During my term as ASME Vice President of Basic Engineering, I had the opportunity to work with AMD Chairs Dusan Krajcinovic, Stelios Kyriakides, Pol Spanos and Mary Boyce.

On the USNC/TAM, I have had the good fortune to work with Tinsley Oden, Jan Achenbach, Bruno Boley, Andy Acrivos, Hassan Aref, Zdenek Bazant, Dan Drucker, George Dvorak, Wolfgang Knauss, and K. Ravi-Chandar, and many others.

Other mechanicians I have known, and on occasion worked with, include: Bernard Budiansky, Mike Carroll, Dick Christensen, Steve Crandall, Jim Dally, Frank Essenburg, Bob McMeeking, Paul Naghdi, Bob Plunkett, Chuck Taylor, Nick Hoff and Sea Nemat-Nasser. I have met Julius Miklowitz, William Prager, Eli Sternberg, Morton Gurtin and Warner Koiter. (Comment on Gurtin giving a lecture at IIT when I was a graduate student. Gurtin is in the audience.)

I hope it is obvious that I feel very lucky to be able to say that I have rubbed elbows with some of the giants of mechanics and the Applied Mechanics Division.

I owe a debt of gratitude to Dan Pletta who hired me at Virginia Tech, Dan Frederick who served as my chairman there for many years, and Ed Starke who hired me at the University of Virginia. And, of course, to Marlene who has always been there, and allowed me to follow what were at times, capricious whims.

In closing, I would like to comment briefly on some beliefs that I have arrived at after 50 years in the field. I believe that the sustainable future of mechanics is in the fact that it is a science. While we in the US tend to think of mechanics as an engineering discipline, if you read the literature on the establishment of the International Union of Theoretical and Applied Mechanics, our field was, and is, clearly considered a science.

There are several Engineering Science and Mechanics departments in the US that have maintained strength and vitality while other mechanics departments without Science in the name have floundered. Advances in engineering, in particular computational engineering, have radically changed the practice of engineering during my professional lifetime. We are now more science oriented in our approach to traditional engineering problems as well as the types of new problems that we are investigating.

Further, it is evident that science receives the lion’s share of research funding in the US. This was brought home to the USNC/TAM last spring when we had presentations from congressional staff members. The message was: if you hope to increase funding for mechanics, emphasize the science of mechanics. Indeed, the USNC/TAM is considering a name change that would include the word science. Your collective input to a more modern name for the US Committee would be most welcome.

In my role as Secretary of the USNC/TAM, I want to encourage all of you to submit proposals to host an IUTAM Symposium. These international symposia are opportunities to bring together an international assemblage of experts in a specific field of mechanics. They can have a major impact on the field as well as your local institution. The US has been somewhat lax in proposing IUTAM symposia recently and I encourage you to consider submitting a proposal. All pertinent information can be found on the USNC/TAM web site at USNCTAM.org. The two-page proposals are due in early January 2006.

Again, I thank all of you for honoring me as you have.

2005 AMD Honors and Awards

YOUNG INVESTIGATOR AWARD
George Haller
For seminal contributions to nonlinear dynamics, including the development of analytic theories for chaos near resonance and for unsteady separation in fluid flows

L. Mahadevan
For significant research contributions to the exploration of nonlinear and nonequilibrium phenomena in continuum mechanics using geometry, analysis and scaling ideas in close conjunction with experiments


APPLIED MECHANICS DIVISION AWARD
Carl T. Herakovich
In recognition of his distinguished contributions to mechanics of fibrous composite materials, and his distinguished service to the mechanics and engineering science community


DANIEL C. DRUCKER MEDAL
Robert L. Taylor
For pioneering contributions to computational solid mechanics and, in particular, for the development of methods and software used worldwide for the calculation of inelastic response of structures


WARNER T. KOITER MEDAL
Raymond W. Ogden
For seminal contributions to nonlinear elasticity, its mathematical foundations and its applications


TIMOSHENKO MEDAL
Grigory I. Barenblatt
For seminal contributions to almost every area of solid and fluid mechanics,including fracture mechanics, turbulence, stratified flows, flames, flow in porous media, and the theory and application of intermediate asymptotics