The High-Order Shifted Boundary Method: A Finite Element Method for Complex Geometries without Boundary-Fitted Grids

High fidelity computational methods are an invaluable tool for analysis, with many breakthroughs in the simulation and understanding of complex physics phenomena. However, over the past two decades, high-fidelity methods have faced the daunting challenge of an increasing geometric complexity of the shapes to be simulated. Additive manufacturing and optimization raised the geometric complexity of designs to new heights, and the current algorithms are lagging behind.

Because of the specific computational infrastructure of a high-fidelity method, setting up the geometrical description of design shapes takes more time than the actual computation. Consequently, high-fidelity computational methods for physics modeling have often been confined to simple design shapes. This project is aimed at breaking this barrier, introducing a new way of computationally model the boundary surfaces of complex geometrical objects.

This project aims to transform the field of computing as we know it, fostering a renaissance of high-fidelity methods in scientific computing, with broad benefits in all fields of science and engineering, including the interface of simulation with artificial intelligence and other meta-algorithms, digital twins, etc.

High-Order Finite Element Methods (HO-FEMs) were originally applied to computational physics problems, with the primary goal of supporting the scientific understanding of complex multi-scale phenomena. Later, HO-FEMs have extended their realm of applications to engineering simulations, in which geometrically complex design shapes are very frequent. In this case, mesh generation with curvilinear elements is necessary to retain optimal accuracy near boundaries.

This task is rather involved, and low levels of automation are often experienced, with a consequent slow-down of the entire design and analysis cycle. In 2018, the Shifted Boundary Method (SBM) was developed as an alternative to traditional methods. In the SBM, which belongs to the broad class of approximate/immersed boundary methods, the location where boundary conditions are applied is shifted from the true boundary to an approximate (surrogate) boundary.

At the same time, the value of boundary conditions, applied weakly, is modified (shifted) by means of Taylor expansions to maintain optimal accuracy. The SBM is a simple, robust, accurate and efficient algorithm for very complex geometries, including the case of non-watertight boundary surfaces. This project aims at developing the higher-order SBM (HO-SBM) and its mathematical analysis of numerical stability and accuracy, for the Poisson, Stokes, Darcy, and compressible Euler equations.

HO-SBM has several advantages: first and foremost, it does not require curved grid edges along the surrogate boundary to obtain optimal accuracy. Complex geometries are characterized by the distance between the surrogate boundary and true boundary of the shapes to be simulated. Hence, the HO-SBM has a flexible integration with current CAD and mesh generation and can help the broad diffusion of reduced-order modeling, machine learning, uncertainty quantification, and optimization methods to complex engineering problems.

Together with the education of a graduate student in computational mathematics and sciences, this projects also aims at attracting undergraduate students interested using computing for design, by exposing them to simplified, easy-to-use versions of the HO-SBM method.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.