Fast Meshless Reanalysis Using Combined Approximations, Preconditioned Conjugate Gradient, and Taylor Series

Abstract

Multiple design iterations often require repeated stress analyses to be performed as the design is modified slightly. A method is presented that combines the meshless stress analysis method with a reanalysis technique to avoid repeating the time-consuming steps of remeshing and solving for small design changes. An iterative reanalysis method based on the preconditioned conjugate gradient method is introduced and compared to the linear Taylor series, simple iteration, and combined approximations reanalysis methods. The asymptotic running time is presented for each reanalysis method, and accuracy is compared for two example problems: a cantilever beam and a hole-in-plate. Results show the Taylor series to have the fastest run time, followed in order by simple iteration, preconditioned conjugate gradient, and combined approximations. For the two example problems, accuracy of the simple iteration method is poor for design changes greater than 5%. Taylor series accuracy depends greatly on the choice of the design variable, the example problem, and the method for computing the sensitivity. The combined approximations and preconditioned conjugate gradient methods both demonstrate less than 10% error up to a 100% change in height for the cantilever beam and 30% change in radius for the hole-in-plate example. Multiple design iterations often require repeated stress analyses to be performed as the design is modified slightly. A method is presented that combines the meshless stress analysis method with a reanalysis technique to avoid repeating the time-consuming steps of remeshing and solving for small design changes. An iterative reanalysis method based on the preconditioned conjugate gradient method is introduced and compared to the linear Taylor series, simple iteration, and combined approximations reanalysis methods. The asymptotic running time is presented for each reanalysis method, and accuracy is compared for two example problems: a cantilever beam and a hole-in-plate. Results show the Taylor series to have the fastest run time, followed in order by simple iteration, preconditioned conjugate gradient, and combined approximations. For the two example problems, accuracy of the simple iteration method is poor for design changes greater than 5%. Taylor series accuracy depends greatly on the choice of the design variable, the example problem, and the method for computing the sensitivity. The combined approximations and preconditioned conjugate gradient methods both demonstrate less than 10% error up to a 100% change in height for the cantilever beam and 30% change in radius for the hole-in-plate example. Disciplines Mechanical Engineering Comments Nomenclature

Virtual Reality Interactive Design Utilizing Meshless Stress Re-Analysis

“…A re-analysis method uses the results from a stress analysis of the original design to quickly compute the stresses for a modified design. Several re-analysis methods were evaluated and the PCG method was found to be the most efficient [23].…”

Haptic Feedback to Guide Interactive Product Design

“…The fast reanalysis uses a preconditioned conjugate gradient (PCG) method to rapidly resolve the system of equations arising from the meshfree analysis. By using the factored stiffness matrix from the previous analysis, the PCG method can quickly solve the system for the deformed part shape [11]. The final result was a combination of Taylor series approximation for interactive analysis and PCG being used once the user stopped changing the part geometry.…”

Interactive Deformation Through Mesh-Free Stress Analysis in Virtual Reality

“…Re-meshing can be very compute intensive and therefore, this VR testbed required an analysis method which could handle large deformations without automated re-meshing. A time comparison between traditional FEA and the mesh-free method was explained in [13].…”

“…In addition, a PCG reanalysis method was added to rapidly and accurately compute the stress contours resulting from larger design changes without recalculating the entire matrix of equations [13]. Chipperfield et al developed a two step process to approximate the stresses.…”