Collaborative research: analysis and computational methods for blended modeling of dynamic and quasistatic fracture

The National Science Foundation Division of Mathematical Sciences awarded Grinnell College $250,000 on September 1, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to develop theory and numerical methods for computing solutions to the fracture initial value problem in continuum mechanics.

The project addresses the computational modeling of dynamic and quasistatic fracture propagation in complex structural geometries under environmental loads. The recipient will formulate a mathematically well-posed initial value problem that predicts damage and fracture patterns by evolving material displacement according to Newton's second law, seeking a displacement-fracture pair informed by initial conditions and time-dependent boundary conditions. The blended modeling approach uses a minimal set of material parameters to predict fracture phenomena, including crack nucleation, and differs from traditional phase-field models by relying solely on the equation of motion for material displacement.

The project includes developing effective computational platforms for GPU and high-performance computing cluster implementation. Undergraduate and graduate students will receive training in active development of numerical methods for these platforms. The recipient plans to offer several courses and coordinate with experimental materials scientists throughout the grant period.

Performance occurs at Grinnell College in Grinnell, Iowa. The period of performance runs from September 1, 2026, through August 31, 2029.

Abstract

The development of theory and numerical methods for computing solutions to the fracture initial value problem remains an underappreciated but outstanding problem of continuum mechanics and mathematics. A predictive theory of fracture and damage propagation capable of handling complex structural geometries and environmental loads would provide an effective tool used for the design of infrastructure, transportation, and energy production. The first goal of the project is to provide an initial value problem useful for prediction of damage and fracture patterns as the outcome of a mathematically well-posed initial value problem that can be implemented on computational platforms. The second goal is to provide effective computational platforms to carry out this task. The project involves training undergraduate and graduate students, who will be exposed to the active development of numerical methods for gpus and high-performance computing clusters. Several courses are planned as are interactions with experimental materials scientists over the course of the grant period. This project contributes to the development of theory and numerical methods for computing solutions to the fracture initial value problem. An evolution for the displacement inside the material based upon newton's second law is to be given and a displacement-fracture pair is sought. The evolution is informed by initial values for the displacements and velocities and is driven by boundary conditions that change in time. Our approach seeks to quantitatively model fracture phenomena using the least number of material parameters to predict fracture patterns, including crack nucleation. Unlike traditional phase field models for fracture, the blended approach proposed here involves only the equation of motion for the material displacement. Similar to classical elasticity, IT is based only on newton s second law and does not require a separate equation for the phase field. 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.- subawards are not planned for this award.