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The following four-article series was published in a newsletter of the American Society of Mechanical Engineers (ASME). It serves as an introduction to the recent analysis discipline known as the finite element method. The author is an engineering consultant and expert witness specializing in finite element analysis. FINITE ELEMENT ANALYSIS: Solution by Steve Roensch, President, Roensch & Associates Third in a four-part series While the pre-processing and post-processing phases of the finite element method are interactive and time-consuming for the analyst, the solution is often a batch process, and is demanding of computer resource. The governing equations are assembled into matrix form and are solved numerically. The assembly process depends not only on the type of analysis (e.g. static or dynamic), but also on the model's element types and properties, material properties and boundary conditions. In the case of a linear static structural analysis, the assembled equation is of the form Kd = r, where K is the system stiffness matrix, d is the nodal degree of freedom (dof) displacement vector, and r is the applied nodal load vector. To appreciate this equation, one must begin with the underlying elasticity theory. The strain-displacement relation may be introduced into the stress-strain relation to express stress in terms of displacement. Under the assumption of compatibility, the differential equations of equilibrium in concert with the boundary conditions then determine a unique displacement field solution, which in turn determines the strain and stress fields. The chances of directly solving these equations are slim to none for anything but the most trivial geometries, hence the need for approximate numerical techniques presents itself. A finite element mesh is actually a displacement-nodal displacement relation, which, through the element interpolation scheme, determines the displacement anywhere in an element given the values of its nodal dof. Introducing this relation into the strain-displacement relation, we may express strain in terms of the nodal displacement, element interpolation scheme and differential operator matrix. Recalling that the expression for the potential energy of an elastic body includes an integral for strain energy stored (dependent upon the strain field) and integrals for work done by external forces (dependent upon the displacement field), we can therefore express system potential energy in terms of nodal displacement. Applying the principle of minimum potential energy, we may set the partial derivative of potential energy with respect to the nodal dof vector to zero, resulting in: a summation of element stiffness integrals, multiplied by the nodal displacement vector, equals a summation of load integrals. Each stiffness integral results in an element stiffness matrix, which sum to produce the system stiffness matrix, and the summation of load integrals yields the applied load vector, resulting in Kd = r. In practice, integration rules are applied to elements, loads appear in the r vector, and nodal dof boundary conditions may appear in the d vector or may be partitioned out of the equation. Solution methods for finite element matrix equations are plentiful. In the case of the linear static Kd = r, inverting K is computationally expensive and numerically unstable. A better technique is Cholesky factorization, a form of Gauss elimination, and a minor variation on the "LDU" factorization theme. The K matrix may be efficiently factored into LDU, where L is lower triangular, D is diagonal, and U is upper triangular, resulting in LDUd = r. Since L and D are easily inverted, and U is upper triangular, d may be determined by back-substitution. Another popular approach is the wavefront method, which assembles and reduces the equations at the same time. Some of the best modern solution methods employ sparse matrix techniques. Because node-to-node stiffnesses are non-zero only for nearby node pairs, the stiffness matrix has a large number of zero entries. This can be exploited to reduce solution time and storage by a factor of 10 or more. Improved solution methods are continually being developed. The key point is that the analyst must understand the solution technique being applied. Dynamic analysis for too many analysts means normal modes. Knowledge of the natural frequencies and mode shapes of a design may be enough in the case of a single-frequency vibration of an existing product or prototype, with FEA being used to investigate the effects of mass, stiffness and damping modifications. When investigating a future product, or an existing design with multiple modes excited, forced response modeling should be used to apply the expected transient or frequency environment to estimate the displacement and even dynamic stress at each time step. This discussion has assumed h-code elements, for which the order of the interpolation polynomials is fixed. Another technique, p-code, increases the order iteratively until convergence, with error estimates available after one analysis. Finally, the boundary element method places elements only along the geometrical boundary. These techniques have limitations, but expect to see more of them in the near future. Next month's article will discuss the post-processing phase of the finite element method. Roensch & Associates. All rights reserved.

Artificial Intelligence as a research field was born in the summer of 1956 during a seminal workshop at Dartmouth College in Hanover, New Hampshire. It was just a year before that when Marvin Minsky, Nathaniel Rochester, Claude Shannon and John McCarthy proposed that they should hold a workshop to put together a roadmap about how to make machines think and learn similarly to humans. The ultimate goal was to discover computational models in order to enable machines to do commonsense reasoning. Today, John McCarthy is rightly considered the father of AI. I should note that the term "Artificial Intelligence" appeared for the first time in the proposal put forth by the previously mentioned scientists. And so this new discipline that would eventually captivate everyone’s imagination was born. Artificial Intelligence had its ups and downs in the last 50 years. Early success solving small problems in simulation ignited a flurry of predictions about super intelligent machines taking over the world before the coming of the 21st century. Hampered by a lack of a good understanding of how commonsense reasoning works in people and a lack of computational resources, computers being very slow up until the mid nineties, AI research stalled in the 80s. Many people rushed to dismiss it as nothing more than hot air. However, science is all about proposing and testing new theories in order to find the best ones. Since the mid-90s, AI research has advanced by leaps and bounds. We now have a better understanding of how the human brain works and that has helped us to find and test better computational models for AI. These in turn have also helped us to better understand the functions of the human brain. New techniques such as statistical analysis are helping intelligent agents to copy with large amounts of information and noisy sensors. Faster computers with vast amounts of storage are allowing us to experiment in more challenging domains and solve larger problems. It is true that AI has not yet been able to produce a machine capable of commonsense reasoning. However, by specialization, many AI systems are actually running our world today. AI helps us fly airplanes and drive our cars. It aids doctors perform surgery. It helps us find information in the vastness of the World Wide Web. It helps us discover spam email and promptly delete it. It helps us schedule traffic lights and public transportation. It helps us analyze financial markets and make predictions about the outcome of sports events. It aids in surveillance of public spaces improving security and safety. These are only a small sample of the penetration of intelligent systems in our daily lives. Artificial Intelligence is here to stay and I bet it won't be long before we have the understanding, methods and resources to finally construct thinking and learning machines. Let us wish and hope that such technology would only be used to benefit mankind and not destroy it. You can find lots of information about AI' and its50th birthday on the Internet. However, I think that best reading about this topic is the 1955 proposal for the AI workshop. You can read it here.