Particle methods for Engineering Mechanics II

Numerical simulation of materials’ large deformation is important for various problems such as slope stability, granular flow dynamics, and impact protection. Particle-based methods (PM) for continuum mechanics could handle large deformation with unprecedented accuracy and robustness. PM is becoming increasingly popular among geotechnical and mechanical engineers.

This course provides students with both in-depth and comprehensive knowledge of three key particle based methods, i.e., material point method (MPM), Smooth Particle hydradynamics (SPH) and Peridynamics (PD). Through the learning process, students will gain a solid understanding of the concepts, derivations and methodological fundamentals of particle-based discretization in continuum mechanics. Meanwhile, students will be trained in enhancing their programming skills with a concentration on data structure optimization and parallelization, which will act as a good supplementary for their learning in other numerical-based courses.

Upon completion of this course, students will be able to develop their own particle-based solvers for computational continuum mechancis and implement several well-established constitutive models for modelling elasto-plastic and visco-plastic behaviours in a range of engineering problems, such as slope failures, granular collapses, and metal deformation. For graduates who are interested in the pursuit of a research career in particle-based methods, this course offers a solid foundation for their future innovations.

• Formulate numerical (finite element) approximations to the equations of motion governing the large, possibly dynamic, deformations of continua.
• Formulate variational update algorithms for the integration of the constitutive equations modeling a wide range of material behavior, including finite elasticity, plasticity and rate-dependency.
• Implement the resulting algorithms in a computer program.
• Apply the computer program to the solution of concrete engineering science and engineering design problems.
• Formulate numerical (finite element) approximations to the equations of motion governing the large, possibly dynamic, deformations of continua.
• Understand the physical and mathematical fundamentals of continuum particle-based methods.
• Formulate variational form and discritization for the integrations of governing PDEs using particle based methods.
• Formulate the numerical algorithm of constitutive models for simulating material behaviours including, hyper-elasticity, elastoplasticity, viscoplasticity, and fractures.
• Design and implement continuum particle-based methods for various applications such as solid mechanics and soil mechanics.

The course offers comprehensive information on the physical origin, mathematical derivation and numerical implementation of material point method (MPM). Students are expected to have basic concepts of continuum mechanics and finite element method (FEM), and familiarity with multi-variable calculus and linear algebra. Detailed contents will be covered as follows:

1. Descriptions of material motion
2. Governing equations, conservation of mass/linear momentum
3. Discretization of MPM
4. MPM algorithm of explicit integration
5. MPM algorithm of implicit integration
6. Discretization of SPH
7. SPH algorithm for explicit integration
8. SPH algorithm for implicit integration
9. Discritization of PD
10. Implement of constitutive models
11. Engineering case studies using MPM (soil mechanics)
12. Engineering case studies using SPH (Fulid mechancis)
13. Engineering case studies using PD (Fracture mechancis)