Parallel splitting solvers for the isogeometric analysis of the Cahn-Hilliard equation

Area of Science:

• Computational biology
• Mathematical modeling
• Cancer research

Background:

• Tumor growth modeling is complex due to dynamic biological and chemical processes.
• Simulating interactions between tumor cells, host tissue, and vasculature is computationally intensive.
• Accurate models are crucial for cancer diagnosis and treatment strategies.

Purpose of the Study:

• To develop an efficient computational model for simulating tumor-host interactions.
• To apply the Cahn-Hilliard phase-field equation for modeling intricate biological processes.
• To leverage advanced numerical methods for improved simulation speed and accuracy.

Main Methods:

• Utilized the Cahn-Hilliard phase-field equation to model tumor-host tissue interactions.
• Employed highly-continuous isogeometric elements for spatial discretization.
• Implemented an alternating directions implicit methodology for efficient time-dependent simulations.
• Reduced computational complexity to Kronecker products of 1D matrices for linear cost factorization.

Main Results:

• Achieved fast simulations of the time-dependent Cahn-Hilliard equation.
• Demonstrated linear computational cost through matrix factorization.
• Enabled parallel multi-core simulations with good scalability on shared-memory systems.
• Showcased high efficiency and accuracy using isogeometric elements on tensor-product meshes.

Conclusions:

• The developed phase-field model provides an efficient and accurate method for simulating tumor growth.
• Isogeometric analysis combined with an implicit time-stepping scheme significantly enhances computational performance.
• This approach offers a powerful tool for advancing cancer research and therapeutic development.