Short Courses

Oscar Lopez-Pamies, University of Illinois Urbana-Champaign

John E. Dolbow, Duke University

Description:

This short course will present the mathematical formulation and the associated numerical implementation of the phase-field approach to fracture. In a nutshell, the phase-field approach to fracture is the culmination of combined efforts (started at then of the 1990s) by the mathematics and mechanics communities aimed at describing where and when fracture nucleates and propagates in solids under arbitrary mechanical loads in a computationally tractable manner. These efforts comprise three pivotal ideas, in chronological order: (i) the casting of the phenomenon of fracture propagation as a variational problem [1], (ii) its regularization into second-order PDEs [2], and (iii) the generalization of these PDEs to account for fracture nucleation at large [3-6]. The latter two ideas constitute the phase-field approach to fracture.

Specifically, the course will focus on the phase-field approach to elastic brittle materials like glass, ceramics, and elastomers. In such materials, the energy is dissipated only through the creation of new surfaces and is proportional to the amount of surface area created. Fracture toughness is the proportionality constant and constitutes one of the three material inputs in the theory. The second material input is the stored-energy function describing the elasticity of the material. The third material input is the strength surface.

The course will include a detailed introduction to the three pivotal ideas listed above, and the constitutive choices that are made to develop a general phase-field model. The casting of the model in a finite element formulation will be discussed, and a live demonstration in Python (using FEniCSx library [7]) and in C++ (using MOOSE [8]) will be given to solve representative initial-boundary-value problems involving fracture nucleation and propagation in both linear elastic and hyperelastic materials. The course material will include lecture notes on the fundamentals of the method in addition to the set of FEniCSx and MOOSE codes that will be used for the live demonstration. Helpful references are listed below.

References:

1. Francfort GA, Marigo JJ (1998) J Mech Phys Solids 46:1319–1342.

2. Bourdin B, Francfort GA, Marigo JJ (2000) J Mech Phys Solids 48:797–826.

3. Kumar A, Francfort GA, Lopez-Pamies O (2018) J Mech Phys Solids 112:523–551.

4. Kumar A, Bourdin B, Francfort GA, Lopez-Pamies O (2020) J Mech Phys Solids 142:104027.

5. Kumar A, Lopez-Pamies O (2020). Theor. Appl. Fract. Mech. 107, 102550.

6. Lopez-Pamies O, Dolbow JE, Francfort GA, Larsen CL (2025). In press.

7. FEniCSx computing platform, https://docs.fenicsproject.org/.

8. MOOSE computing platform, https://mooseframework.inl.gov/.

Syllabus:

Hours 1-3: The Theory

• A summary of macroscopic experimental observations of fracture nucleation and propagation in nominally elastic brittle materials

• The definition of strength

• Griffith postulate for fracture propagation as a variational problem

• The phase-field regularization of the Griffith variational problem 

• Accounting for strength to construct a complete phase-field theory of fracture nucleation and propagation

Hours 4-6: The Numerical Implementation

• Weak form and finite element formulation of the governing PDEs

• Staggered scheme to solve the resulting discretized equations

• Representative initial-boundary-value problems: 

  • • Nucleation of fracture under uniaxial tension

    • Nucleation of fracture from a V-notch

    • Propagation of fracture in a pure-shear test

    • Indentation of glass with a cylindrical indenter

    • The Brazilian fracture test for mortar

    • The poker-chip experiment for rubber

Pablo Seleson, Oak Ridge National Laboratory 

John Foster, The University of Texas at Austin

David Littlewood, Sandia National Laboratories

Description

Peridynamics is a nonlocal reformulation of classical continuum mechanics, based on integral equations, suitable for material failure and damage simulation. In contrast to classical constitutive relations, peridynamic models do not require spatial differentiability assumptions of displacement fields, leading to a natural representation of material discontinuities such as cracks. Furthermore, peridynamic models possess length scales, making them suitable for multiscale modeling. This course will provide an overview of peridynamics, including its mathematical, computational, and modeling aspects. The course will also review advanced research topics and software in peridynamics, and it will include a hands-on tutorial on 3D simulation of solid mechanics problems.

Outline

Introduction to Peridynamics 9:00-9:45

Break 9:45-10:00

Peridynamic Material Models 10:00-10:45

Break 10:45-11:00

Computational Peridynamics 11:00-12:00

Lunch 12:00-13:00

Modeling Failure and Damage 13:00-13:30

Multiphysics Modeling in Peridynamics 13:30-14:00

Multiscale Modeling in Peridynamics 14:00-14:30

Break 14:30-14:45

2D Computations in Peridynamics with MATLAB: PDMATLAB2D 14:45-15:30

Practice Session 15:30-16:00

Hands-on Peridigm Tutorial 16:00-17:00

Gary (Tianchen) Hu, Argonne National Laboratory

Mark Messner, Argonne National Laboratory

Overview:

This short course provides a high-level introduction to the NEML2 constitutive modeling library developed by Argonne National Laboratory. The course combines lectures with hands-on tutorials to explore the use and applications of NEML2 in various engineering contexts. Attendees will gain practical skills in utilizing NEML2 for material modeling for both conventional and modern tasks.

Resources:

• Repository: https://github.com/applied-material-modeling/neml2

• Documentation: https://applied-material-modeling.github.io/neml2

• Report/manual: https://www.osti.gov/biblio/2440430

Pre-Requisites:

• Basic understanding of Python and PyTorch.

• Basic C++ skills.

• Familiarity with numerical modeling and engineering mechanics is beneficial but not required.

Session Breakdown:

Part 1: Foundations (1 hr)

• Overview of Constitutive Modeling and NEML2

• Hands-On Session: Setting Up NEML2

Part 2: Model Development (2 hr)

• Modular Constitutive Model Composition

• Flexible Batching and Vectorization Mechanisms

• Hands-On Session: Building Modular Models

Part 3: Advanced Techniques (1.5 hr)

• Automatic Differentiation in NEML2

• Machine Learning in Constitutive Modeling

• Hands-On Session: Developing ML-Integrated Models

Part 4: Real-World Applications (2.5 hr)

• Parameter Calibration via Stochastic Variational Inference

• Coupling with MOOSE for Multiphysics Simulations

• Inverse Optimization in MOOSE-NEML2 Coupling

• Hands-On Session: Parameter Calibration & MOOSE Coupling

Part 5: Summary and Future Directions (1 hr)

• Applications and Open Research Questions

• Panel/Q&A Session

Krishna Garikipati, University of Southern California

Benjamin Jasperson, University of Southern California

Rahul Gulati, University of Southern California

Description:

Large language models (LLMs), already revolutionizing generative text applications, have great potential in many areas of science. In this short course, you will learn about a recent application of LLMs to mechanics-based research and teaching. You will learn how to fine-tune open source LLMs to serve many different science research roles, including spatiotemporal solver and virtual research/teaching assistant.

We will start with a general introduction to the transformer architecture, then proceed to fine-tuning LLMs through low-rank adaptation (LoRA) and adding context-specific reference material through retrieval-augmented generation (RAG). In this process, we will also get familiar with the HuggingFace platform and the libraries used for inference using an LLM.

Next, we highlight recent work where we will dive into the architecture of a Vision Transformer, and use LLMs to solve multiphysics problems using spatiotemporal data. We will close by applying these methods for hyperparameter optimization and to fine-tune an LLM to serve as an AI research/teaching assistant (AI-RA or AI-TA) using material from a finite element course, including an automated approach to training data generation from reference text.

Participants will have a chance to apply these methods to example single/multiphysics problems and/or a course with supplied material. Alternatively, they can bring their own spatiotemporal data from a computational solution or literature/course material to use. All code will be made available to participants after the session. It is expected that participants will be able to take the methods used and immediately apply them to their own research.

Syllabus:

1. Introduction

1.1. LLMs: a review. History, critical literature

1.2. Background: transformer architecture, retrieval-augmented generation (RAG), low-rank adaptation (LoRA)

2. Fine-tune LLM for content-specific materials

2.1. Workbook setup, login, GitHub repo

2.2. Generating training data from text using an LLM: prompt engineering, generating q/a, assessing quality, getting familiar with API calls.

2.3. Implementing RAG for context-specific materials.

3. LLMs for spatiotemporal analysis - hands-on demo

3.1. Background info- extensions to multimodal

3.2. Architecture of ViT / Multimodal VLMs

3.3. Example/working demo

4. Fine-tune LLM for course-specific material (AI-TA) - hands on demo for finite element course

4.1. HuggingFace open source LLM-leaderboard. Inference using an open source LLM

4.2. Hyperparameter optimization. Fine-tune and evaluate an open-source LLM using LoRA: chat templating.

4.3. Working session, Q/A