# The boundary element method extends the limits of analysis.

The boundary element method of analysis is experiencing a resurgence almost a century after the mathematics behind it were first developed. While BEM has been used for decades to analyze complex models in applications such as fracture mechanics, it has largely been ignored commercially. Some software developers have rediscovered BEM and are mounting a challenge to the finite element method.In analysis methodology with its roots in early 20th century field theory is once again at the forefront of research. Universities, government agencies, manufacturers, and software vendors are probing the potential of the boundary element method (BEM) of analysis. Although its origins predate the finite element method (FEM), BEM has experienced stunted commercial development, in part because of its computationally intensive solvers. However, improvements in processor performance have permitted boundary elements to stage a comeback in the 1990s.

Boundary element software is becoming popular in the analysis of complex 3-D solid components, where it is difficult for users to build FEM models even using p-type elements, said Jon Trevelyan, vice president of Computational Mechanics Inc. (Billerica, Mass.), makers of the Beasy BEM code. Detailed features of solid models or models with large volumes can require a large number of tetrahedral finite elements, which can become impractical to create. Boundary elements remove the need to mesh throughout the volume of the component.

Generally, FEM can be applied to a larger range of problems and has been the subject of more product development than BEM. As a result, FEM has emerged as the dominant analysis method for mechanical engineering. However, because FEM has already been so thoroughly explored, many researchers now view BEM as new territory. According to Prasanta Banerjee, professor of civil engineering at the State University of New York (SUNY) in Buffalo, and other researchers, more gains can be expected from studying boundary than finite elements. In fact, some of BEM's traditional shortfalls are being overcome as a result of this research. Similarly, many software developers, such as Cadkey (Windsor, Conn.) and PDA Engineering (Costa Mesa, Calif.), are interested in BEM because it opens doors to markets for analysis products that were previously closed to them.

A Boundary Underground

Fracture mechanics is one area where many engineers prefer BEM to FEM. Beasy, one of the original BEM codes, is capable of modeling highstress gradients at crack tips. The results from a Beasy analysis are consistently within 0.5 percent of FEM solutions. Furthermore, BEM problems can be formulated much faster than comparable FEM problems.

According to Thomas Cruse, professor of mechanical engineering at Vanderbilt University (Nashville, Tenn.) and a pioneer developer of BEM codes, one of the first numerical solutions of a 3-D fracture mechanical problem was derived using BEM. In 1973, Cruse went to work at Pratt & Whitney (East Hartford, Conn.), one of the few companies with a long-standing interest in BEM formulation. Most of the work at Pratt & Whitney was directed at analyzing fracture mechanics in aircraft engines.

"BEM constituted about 10 percent of analysis computations performed at Pratt," Cruse said. "We were able to achieve accuracies for load-stress problems that were better than standard FEA."

One reason Cruse cited for the greater accuracy in certain volume problems is that BEM is less sensitive to mesh distortions than is FEM. Each element, or more properly, cell, is described as a high-level algebraic function. In fact, the real efficiency of BEM was lost if an engineer attempted to "overmesh" by applying a FEM meshing strategy. The more sparingly an engineer built the mesh, the better the results were.

In 1982, NASA and Pratt & Whitney jointly supported a BEM research project conducted at SUNY Buffalo. Other funding was provided by automakers in Detroit and Germany and several other major manufacturers. The efforts have resulted in several public-domain BEM codes released over the last few years, including the Boundary Element Software Technology (BEST) code.

Boundary element codes can carry out linear-elastic stress analyses on regular geometries with great accuracy, said SUNY's Banerjee. The method requires more algebra, but the models are much easier to build because they contain far fewer elements than do finite element models.

Banerjee is also director of the Boundary Element Software Technology Corp. (Getzville, N.Y.), formed to commercialize the BEST code. The product, GPBest (generalpurpose BEST) is a BEM solver written in FORTRAN 77 for IBM mainframes, Cray supercomputers, and a number of Unix workstations. The program currently can handle stress, vibration, heat transfer, acoustic, and fluid flow analyses.

Instead of developing a graphical interface from scratch, Banerjee selected Patran from PDA Engineering as a pre- and post-processor. The PatGPB data-preparation interface takes the neutral file from the Patran work session and translates the nodal coordinates and connectivities of the model into a form usable by GPBest. PatGPB also works as a reverse translator for exporting GP Best results to Patran for post-processing and visualization.

Easier Analysis

Market research by several CAD/CAE software vendors indicates that only a handful of design engineers use any computer analysis products. Traditionally, analysis has been performed by specialists, although some software developers are making inroads with FEA products using the p-method. The often-cited reason is the learning curve--real and imagined--associated with analysis products. In order to get beneficial results from FEA, the model has to be divided into elements by means of a mesh. Generating a mesh that accurately characterizes what is being modeled requires training, practice, and time. The difficulties are compounded in 3-D problems.

The premise of BEM, however, asserts that a volume integral can be reduced to a surface integral, and by calculating the boundary of the volume, the behavior of the interior can be predicted. Under the boundary element method, only the boundary of the problem domain is discretized, which reduces the problem's dimensionality by one. Thus, 3-D volume problems are solved as 2-D surface problems and area problems are modeled as 1-Degrees. The reduction of a problem's dimensionality results in fewer equations being solved using BEM for a given problem compared to FEM. However, each BEM equation will have a larger number of algebraic functions than a corresponding FEM problem. Thus, solving BEM formulations requires more processing power.

In the integral equations solved by BEM, each nodal point is compared with every other nodal point on the model, producing fully populated asymmetric matrices. By contrast, FEM is a differential technique in which each nodal point is calculated only from its neighboring elements, producing symmetrical banded matrices.

Boundary element analysis often is described as "meshless" by its proponents. While this is not strictly true, the user can be insulated from the meshing process because it is far simpler under BEM and lends itself to automated techniques. BEM accuracy stems from the complexity of the formulated equations, not from the complexity of the mesh. This makes BEM analysis products attractive to users who do not care to learn how to mesh.

Design engineers at Northern Telecom's wire and cable plant in Kingston, Ontario, have been evaluating the BEM-based Cadkey Analysis module from Cadkey for several months to analyze mold designs. The plant produces rubber dies and molds for the manufacture of insulating material for telecommunication lines.

According to John Garner, senior staff engineer at the Kingston plant, the major attraction of the Cadkey Analysis product is its simplicity. "It doesn't require learning time. You don't have to spend hours sorting out the mesh," he said.

Garner is quick to point out, however, that analysis requirements at the plant are not particularly stringent. It is not necessary to model a rubber die in great detail in order to acquire a good overall view of how it will behave under load. However, Garner said that boundary element analysis is useful to simulate the characteristics of deflections in rectangular solid bodies to analyze design concepts. "It gives you a gut feeling for how it will operate," he said. "We are reluctant to do this in FEA because of the difficulties involved."

The ability to avoid meshing the domain of a volume problem is the driving force behind BEM's insurgency into FEM's world, said Anil Gupta, the product developer at Cadkey responsible for the BEM solver for Cadkey Analysis. The Cadkey Analysis package can be used to perform thermal, elastic, and thermo-elastic analyses on CAD geometries. Cadkey is positioning the system as a complement to its own Cadkey design system.

"For linear analysis problems of stress and displacement, BEM has no competition," Gupta said. The boundary approach is well-suited to handling linear-elastic stress-strain and steady-state heat-strain problems because they are volume problems. "Not only does it produce more accurate results than FEA, but it is easier to formulate the problem," Gupta said.

Ease of formulation is also in the minds of developers at PDA, who have been working for a number of years on the development of its socalled Trimmed-Element Analysis Method (TEAM) product using BEM to perform analysis on CAD geometries. Another "meshless" analysis product, TEAM can formulate boundary integrals as boundary elements using the original CAD geometry. The TEAM user specifies areas of interest, and the system automatically assigns the correct number of functions necessary to solve the problems within given tolerance levels.

The TEAM product is intended for use by design engineers, not experienced analysts. Hayden Hamilton, product manager at PDA, said TEAM is intended to help design engineers achieve a preliminary understanding of how a design behaves from geometry that already exists. BEM addresses the ease-of-use stumbling block of FEM. PDA said it has attracted a number of partners willing to incorporate TEAM into their CAD/CAM products, including EDS (Maryland Heights, Mo.) for Unigraphics and Matra Datavision (Tewksbury, Mass.) for Euclid-IS.

On the other hand, Hamilton notes, BEM is a fundamentally different technology than FEM and therefore has different practical applications. The very characteristics that make BEM suitable to linear solid analysis problems make it unsuitable for obtaining accurate results from many nonlinear and shell-body problems. "Boundary elements work best when analyzing potato-bodies," Hamilton said. "They assume isotropic composition and a high volume-to-surface ratio."

Other BEM codes are becoming commercially available. For example, Automated Analysis Corp. (Ann Arbor, Mich.) recently introduced a BEM-based analysis program called Comet/BEA for solving 3-D linear i problems for elasticity, heat transfer, and thermo-elasticity.

Software vendors, traditionally noted for their FEM systems, have found occasion to use BEM technology where appropriate. According to Ashrif Ali, a development engineer at Swanson Analysis Systems Inc. (Houston, Pa.), BEM is particularly powerful for calculating field problems for which the problem domain extends to infinity, such as in acoustical and electromagnetic field problems. In these, the field of influence theoretically extends to infinity, although the area of interest is usually bounded. Ali said the standard FEM techniques are inefficient for these problems because of the large number of elements required to formulate them.

Expanding the Boundaries

Since BEM is a boundary solution technique, it can provide boundary conditions for a FEM mesh. Swanson Analysis uses BEM in Ansys to obtain accurate boundary conditions for finite element models of infinite domain problems. However, Ansys cannot calculate boundary elements with its finite element solver. Since FEM discretizes the entire domain and requires that problems be formulated this way, results between BEM and FEM are not directly translatable.

While BEM and FEM data are radically different from one another, some see the two methods existing side by side. Their very difference, in fact, can itself be used as a tool to confirm the accuracy of an analysis model. SUNY Buffalo's Banerjee suggests that similar results from a BEM and a FEM analysis of the same model would demonstrate how close to the correct answer a user was coming in his or her basic assumptions. Similar results arrived at through completely different methods would tend to indicate that the basic assumptions were correct. "Problems people solve today are very complex," Banerjee said. "Performing another analysis with a different method would help engineers sleep better at night."

The two methods can also be used in an integrated way. The results of one method of analysis can be used to help formulate a related analysis using the other method. Coupling BEM and FEM codes in this way could lead to more accurate models, with each method playing to its strengths in solving a complex problem.

"The interaction of linear and nonlinear problems is an area where both BEM and FEM can be brought to bear together," Vanderbilt's Cruse said. "For example, acoustic radiation by submarines could be analyzed as a coupled problem. Boundary elements could be used to represent the infinite domain of the water, and finite elements to represent the submarine."

Cadkey's Gupta added that a coupled approach might be appropriate for classical problems in aerodynamics. The interaction at the boundary layer between an airfoil and the infinite domain of the air could be modeled using both boundary and finite elements.

One of the shortfalls of boundary elements, however, is its weakness in solving nonlinear problems. Timedependent phenomena such as wave propagation and vibration in a volume cannot be represented on the boundary completely. Therefore, BEM cannot be used to model these nonlinear cases with any degree of accuracy.

Perhaps a more severe weakness is the difficulty in modeling thin-shell structures using BEM. While BEM is capable of handling thickness ratios of 10:l--and in some cases up to 25:1-- it generally does not produce accurate results for thin plates and shells. Cruse, in fact, considers attempting to model a thin plate using a 3-D boundary element formulation a misuse of the technology. "Clearly, there are classes of problems where BEM and FEM should be separated," he said.

Meshless Mechanical Engineering

Some researchers are skeptical about the future of BEM. KlausJurgen Bathe, professor of mechanical engineering at MIT (Cambridge, Mass.), observed that while BEM does quite well when applied to linear heat transfer, and nearly as well in linear 3-D elasticity problems, it does not perform well with frametype structures. Bathe, developer of the Automatic Dynamic Incremental Nonlinear Analysis (ADINA) FEM code, said a significant portion of current research work is concerned with nonlinear analysis, an area where BEM does not measure up to FEM.

"Although a lot of research into boundary elements is now being done, the shortcomings of BEM are so great that I am not convinced the method is competitive in industry," Bathe said. "And I would have no hesitation in picking up BEM for ADINA if I saw promise."

Bathe added that the data preparation advantages gained from transforming a volume integral to a surface integral are lost in the resulting volume of algebraic equations that need to be solved. "You pay the price for not meshing the interior by having asymmetric ill-conditioned equations that you have to solve," Bathe said. "Symmetric equations produced in FEM can be processed faster than in BEM by a factor of 10."

Developers of some BEM products consider the benefits of an easy-tomesh analysis method using CAD geometry to be worth BEM's computational requirements. Bathe argues that if the idea is to integrate CAD with analysis, current research in FEM preprocessing is already producing results in this area. Some makers of finite element analysis products for PCs and workstations, notably Rasna Corp. (San Jose, Calif.) and Algor Inc. (Pittsburgh), have developed relatively easy-to-learn meshing schemes. "In five years, there will no longer be a need to mesh," Bathe predicted. "Solutions will exist in FEM where the user never sees a mesh."

Despite such criticism, Bathe considers efforts to explore boundary elements to be worthwhile, if only for the research value. "Maybe some bright person will find something we haven't thought of yet," he mused.

SUNY Buffalo's Banerjee is convinced that further research will prove that BEM is capable of solving any problem that FEM can solve, including analysis of thin-shell structures. The stumbling block is that the formulation of certain problems simply has not been done properly. "Instead of using differential equations of shell theory, we need to apply the theory of reduced formulation to shells as has been done with solids."

Proponents and skeptics agree that there is much left to learn about BEM. Whether or not BEM will be able to achieve the commercial success enjoyed by FEM will depend on whether problems related to its computationally intensive nature and the limitations of its algorithms can be overcome. "FEM has been developed to where there is little further development in theory," Cadkey's Gupta asserted. "BEM research offers the promise of alternative analysis techniques."

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Title Annotation: | engineering methodology |
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Author: | Puttre, Michael |

Publication: | Mechanical Engineering-CIME |

Date: | Jan 1, 1993 |

Words: | 2748 |

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