A Conforming Reproducing Kernel Framework for Agile Simulation of Problems with Complex Geometries
Computational mechanics is a growing discipline that is commonly used to provide invaluable insight for engineering analyses, particularly when theoretical or empirical analyses are impractical. Computer simulations can handle complex problems more readily than analytical methods and are often much less expensive and less time consuming than performing experiments. However, developing simulation models for moderate to complex geometries is still a time consuming task. The upfront cost of developing models is an impediment that reduces the effectiveness of simulation in engineering design and analysis.
Meshfree methods provide flexible means of creating approximation functions using only a collection of points with corresponding window functions. This flexibility makes meshfree methods attractive for problems involving large deformation and failure, where mesh-based methods, such as the finite element method, experience complications due to mesh entanglement. Without the need of generating a quality mesh, meshfree methods also have a greatly simplified simulation development process. However, for problems with complicated geometries such as non-convex or multi-body domains, poor solution accuracy and reduced convergence rates are observed due to inaccurate representation of complex geometry unless the extents of the window functions, and thus approximation functions, are carefully controlled. The required control is not available with the Euclidean metrics that are commonly used to construct the window functions.
In this dissertation, methods of providing more control in window function design are presented. ``Conforming'' window functions are constructed using local triangulations in two and three dimensions, allowing efficient and systematic handling of complex geometries. Graph distances are used in conjunction with Euclidean metrics to provide adequate information for shaping the window functions. The conforming window functions are demonstrated using the Reproducing Kernel Particle Method showing improved accuracy and convergence rates for problems with challenging geometries. Additional attention is given to addressing the challenges of simulating nearly incompressible material response and to maintaining a higher stable time increment for dynamic problems that are solved using explicit time integration.
Addressing the boundary-related challenges of meshfree methods both opens the methods to a broader class of problems and enables an agile simulation development process.