An illustration of analytical/numerical matching with finite-element analysis for structural vibration problems

Analytical/numerical matching (ANM) is an accurate and efficient method for solving many types of problems with discontinuities. The method separates local and global effects, and solves separate subproblems using high resolution around the discontinuity and low resolution away from the discontinuity. The work presented in this manuscript demonstrates a methodology for applying ANM to a dynamic structure using finite-element analysis (FEA) for the solution of the high-resolution (local) and the low-resolution (global) subproblems. The ANM method is illustrated on a thick, two-dimensional beam having several displacement constraints attached to its lower surface. Ordinarily (and here, for verification purposes) this problem would be solved using two-dimensional plane elements due to the local discontinuities around the constraints and the thickness of the beam. Using ANM, these discontinuities and through-thickness effects are modeled in the geometrically compact local problem using a high-resolution mesh of two-dimensional eight-node plane elements. The much larger global problem contains no discontinuities and is reduced to the solution of a low-resolution finite-element mesh of two-node Bernoulli–Euler beam elements. A third subproblem (matching) is solved analytically (no computational overhead). The agreement between the ANM solution and the purely FEA solution is excellent, and the computational savings are significant.