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:

9:00 – 9:45: Introduction to Peridynamics

9:45 – 10:00: Break

10:00 – 10:45: Peridynamic Material Models

10:45 – 11:00: Break

11:00 – 12:00: Computational Peridynamics

12:00 – 1:00: Lunch

1:00 – 1:30: Modeling Failure and Damage

1:30 – 2:00: Multiphysics Modeling in Peridynamics

2:00 – 2:30: Multiscale Modeling in Peridynamics

2:30 – 2:45: Break

2:45 – 3:30: 2D Computations in Peridynamics with MATLAB: PDMATLAB2D

3:30 – 4:00: Practice Session

4:00 – 5:00: Hands-on Peridigm Tutorial

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

Bora Pulatsu, Carleton University

Objectives:

• Learning the essential concepts and numerical procedures related to the discrete element method.

• Understanding the block-based representation of load-bearing structural systems. 

• Implementing different contact constitutive laws and understanding their influence on the macro-level.

• Performing linear and non-linear analysis of various masonry structures using discrete element method. 

Description:

Computational modeling of masonry structures can be grouped under continuum and discontinuum-based analysis. The former approach represents masonry composite as a continuous medium with no distinction between masonry constituents, while the latter explicitly considers masonry units and their interaction within the adopted discrete-block modeling framework. The course covers discontinuum-based analysis of masonry structures based on the discrete element method (DEM) and comprehensively discusses the essential features/concepts of DEM. The emphasis is given to the explicit solution scheme of DEM, rigid-body dynamics, mechanical interactions in multi-block discontinuous systems, and contact mechanics. Furthermore, a fundamental background related to numerical methods will be presented. An experiential learning environment will be offered through the implementation of hands-on DEM applications. Structural analysis of unreinforced masonry arches, walls, and large-scale buildings will be performed using a step-by-step approach to discuss selected problems. The proposed course aims to provide a solid background in DEM and discontinuum-based analysis of URM structures containing state-of-the-art applications and course materials.

Subjects that will be covered during the short course (6 hours)

Part I (2 hours) – Fundamentals

Finite difference method (implementation, discretization, stability, and accuracy) 

Dynamic relaxation and Cundall’s local damping approach

Hands-on application of explicit finite difference method.

Part-II (2 hours) – Discrete Element Method (DEM)

Historical background of block-based simulation of a discontinuous medium.

Computational procedure of DEM (illustration on a simple multi-rigid block system).

Mechanical interaction among the adjacent rigid blocks and contact mechanics.

Discussion on the various contact constitutive models and their implementation in the DEM procedure. 

Part-III (2 hours) – Applications 

Representation of discontinuum via a system of rigid blocks

Structural analysis of various load-bearing masonry structures (arch, vault, walls, and large-scale building) using DEM

Discuss the salient features of discrete rigid block simulations considering different engineering problems. 

Course Materials

All lecture notes and codes prepared by the instructor will be available to the participants.

Christopher Eldred, Sandia National Laboratories

Artur Palha, Delft University of Technology

Description:

Systems of partial differential equations (PDEs) underlie most of the models used in the study of continuum mechanics (CMMs), across a range of disciplines such as climate, aerospace, materials science, biomedicine, porous media, combustion and plasma dynamics. Since CMMs are generally intractable by hand, especially for complex multiphysics applications, they must be simulated using numerical methods instead. In doing so, stable and physically accurate simulations are needed. A powerful way to achieve this is through the use of geometric mechanics (GM) formulations (variational, Hamiltonian, metriplectic, etc.) combined with structure-preserving (SP) discretizations. In this approach, the equations of motion are first written in terms of a GM formulation, and then the GM formulation itself (rather than the equations of motion) is discretized using a SP discretization. By doing, so a discrete version of the GM formulation can be developed, with many of the same key properties as the continuous one. This includes conservation laws, involution constraints and freedom from spurious (computational) modes.

This course will serve as an introduction to these ideas, illustrating the entire GM/SP process for the shallow water equations, a prototypical example of a continuum mechanical system. It will consist of a combination of lectures and hands-on work, with the latter consisting of both coding and pen+paper derivations. Longer course notes will be made available online, that discuss the general theory in detail and provide many examples of continuum mechanics systems that can be treated using the GM/SP approach. A companion session (0306: Geometric mechanics formulations and structure-preserving discretizations for continuum mechanics and kinetic models) will happen during the main WCCM conference.

Prerequisites: vector calculus, basis knowledge of PDEs, basic knowledge of finite element methods

Syllabus:

8:30 - 10:00 Intro / Motivating Example

What are geometric mechanics formulations? What are structure-preserving discretizations? Why do they matter? How are they applied In this lecture we will discuss answers to these questions, along with a motivating example: the shallow water equations.

10:00 - 10:15 Coffee Break

Drink a coffee, have a snack, contemplate the meaning of the universe...

10:15 - 12:00 Geometric Mechanics Formulations

In this lecture, we will give an introduction to Geometric Mechanics, focusing on basic geometric mechanics formulations (variational and Hamiltonian) for the shallow water equations. Additionally, there will be some brief discussion of more advanced topics such as formulations for (thermodynamically) irreversible processes (ex. metriplectic); and the semi-direct product theory and exterior calculus that underlies geometric mechanics formulations.

12:00 - 1:00 Lunch Break

Sample the delicious food available in Chicago!

1:00 - 2:30 Structure-Preserving Discretizations I

In this lecture, we will cover the basic theory of structure-preserving discretizations, concentrating on the core concept of the de Rham complex, and classification of fields that appear in physical field theories.

2:30 - 2:45 Coffee Break

Drink a coffee, have a snack, contemplate the meaning of the universe...

2:45 - 4:15 Structure-Preserving Discretizations II

In this lecture, we will continue the discussion of structure-preserving discretizations, focusing on finite-element type methods (ex. finite element exterior calculus) in the context of the shallow water equations. Additionally, there will be some brief discussion of more advanced topics such as single & double de Rham complex methods, variational integrators and structure-preserving time discretization.

4:15 - 4:30 Wrap Up

We will wrap up the course with discussion about how things went, and point out additional more detailed material that is available online.

Martin van der Schelling, Delft University of Technology

Miguel Bessa, Brown University

Description:

The importance of Machine Learning in Computational Mechanics has dramatically grown in the last 5 years. Despite impressive progress, replicating our community's data-driven research results remains a challenge because we lack open-source and user-friendly frameworks. This short-course is focused on an overview of the data-driven process in Computational Mechanics and the corresponding open-source Framework for Data-Driven Design & Analysis of Structures & Materials (f3dasm) [1].

The framework integrates (1) design-of-experiments, where input features describe the microstructure, properties and external conditions of the system, (2) computational analyses, where a material response database is created, and (3) machine learning and optimization, where we either train a surrogate model to fit our experimental findings or iteratively improve the model to obtain a new design.

At the end of this short-course you will be able to replicate the results of a couple of research articles from the literature and, more importantly, be able to use different tools to perform new data-driven investigations to pursue your own research endeavors. Therefore, the learning objectives are:

• reviewing the data-driven process for computational mechanics;

• learning how to use f3dasm with the methods that are already implemented by following in-

• class tutorials;

• learning how to contribute with new methods for future projects of your interest; hence,

• contributing to the open-source project.

The f3dasm framework, the latest syllabus and content of the short-course is available on the f3dasm GitHub page (https://github.com/bessagroup/f3dasm).

Program:

1h30 Part 1: Introduction to the f3dasm framework

Brief introduction to data-driven design for modeling of materials and structures.

Introduction to the f3dasm framework: design-of-experiments, simulation, machine learning and optimization.

15 min Break

1h30 Part 2: Practical session – fundamentals of the framework

Learn how to get familiar with the f3dasm using the design-of-experiments, machine learning and optimization sub-modules.

Hands-on exercises of establishing a machine learning model based on benchmark functions.

60 min Break

1h30 Part 3: Setting up a computational experiment design

Learn to use the framework to perform new data-driven investigations to pursue your own research endeavor.

Illustrative example of a case study: super-compressible material design [2].

Learn how to contribute with new methods to the open-source project.

15 min Break

1h30 Part 4: Practical session – case studies on ongoing work from the Bessa group

An interactive hands-on session where participants can explore a variety of ongoing research directions from the Bessa research group , including topology optimization, constitutive modeling, meta-optimization, and graph neural networks.

Closing Remarks

Programming: 

Tutorials are in Python. Tutorials will be done with the help of Google Colab, therefore no installation is required other than having a Google account.

References:

[1] van der Schelling, M. P., B. P. Ferreira, and M. A. Bessa. "f3dasm: Framework for data-driven design and analysis of structures and materials." Journal of Open Source Software 9.100 (2024): 6912. https://joss.theoj.org/papers/10.21105/joss.06912

[2] Bessa, M. A., Glowacki, P., & Houlder, M. (2019). Bayesian Machine Learning in Metamaterial Design: Fragile Becomes Supercompressible. Advanced Materials, 31(48), 1–6. https://doi.org/10.1002/adma.201904845

Ellen Kuhl, Stanford University

Skyler St Pierre, Stanford University

Kevin Linka, Hamburg University of Technology

Mathias Peirlinck, Delft University of Technology

Description:

In this short course, you will learn about automated model discovery; participate in a hands-on programming experience to implement and train your own Constitutive Artificial Neural Networks; and receive a library of Neural Network notebooks to analyze and interpret classical benchmark data of man-made materials like rubber and living materials like the human brain or skin.

For more than 100 years, chemical, physical, and material scientists have proposed competing constitutive models to best characterize the behavior of man-made and natural materials in response to mechanical loading. Now, computer science offers a universal solution: neural networks. Neural networks are powerful function approximators that learn constitutive relations from big data without any knowledge of the underlying physics. However, classical neural networks entirely ignore a century of research in constitutive modeling, violate thermodynamic considerations, and fail to predict the behavior outside the training regime. In this short course, we introduce Constitutive Artificial Neural Networks, a new family of neural networks that inherently satisfy common kinematic, thermodynamic, and physical constraints and, at the same time, constrain the design space of admissible functions to create robust approximators, even in the presence of sparse data. We revisit the non-linear field theories of mechanics and reverse-engineer the network input to account for material objectivity, material symmetry and incompressibility; the network output to enforce thermodynamic consistency; the activation functions to implement physically reasonable restrictions; and the network architecture to ensure polyconvexity. We show that this new class of network models is a generalization of the classical neo Hooke, Blatz Ko, Mooney Rivlin, Yeoh, Demiray, and Holzapfel models and that the network weights have a clear physical interpretation in the form of shear moduli, stiffness-like parameters, and exponential coefficients.

To familiarize yourself with this new technology, you will implement your own Constitutive Artificial Neural Network and train it with classical benchmark data, for example, from rubber, brain, and skin. You are welcome to bring your own data! You will see that your Constitutive Artificial Neural Network autonomously selects the best constitutive model, parameters, and experiment to characterize your material. This technology could have the potential to induce a paradigm shift in constitutive modeling, from user-defined model selection to automated model discovery. At the end of the course, we will share source codes, benchmark data, and documented examples.

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

N. Sukumar, University of California, Davis

Gianmarco Manzini, Los Alamos National Laboratory

Description:

In this short course, we will present the foundations and applications of the Virtual Element Method (VEM) [1, 2] in solid and fluid mechanics. The VEM is a stabilized Galerkin formulation that permits robust and accurate computations on arbitrary (convex and nonconvex) polygonal and polyhedral meshes. It provides a variational framework for mimetic finite differences and also generalizes hourglass finite elements to arbitrary polytopal meshes. Over the past decade it has become the subject of substantial research and new formulations of the method have appeared to solve initial/boundary-value problems in solid and fluid continua. The VEM afford significant flexibility in element geometries that are permissible: for example, nonconvex elements, elements with short edges in 2D and small faces in 3D, and hanging nodes in a mesh to name a few. In addition, it provides new opportunities to enable accurate and robust computations for finite elements on poor-quality finite element meshes and to treat incompressibility in solid and fluid continua. This short course will be beneficial to graduate student researchers, scientists and academic faculty to gain expertise in this emerging method in computational mechanics. The course will contain 5 lectures and a hands-on two-hour tutorial session. For the hands-on tutorial session, participants should bring a laptop with an installation of Matlab. The course outline is as follows:

Topics Covered in Lectures:

1. Introduction to VEM: Introduction to polytopal computations and the conforming virtual element method, drawing on its connections to hourglass finite elements. Accurate and efficient Numerical integration of polynomials and nonpolynomial functions over polytopes. (Sukumar)

2. First-order (k = 1) VEM for the Poisson problem in 2D and 3D: Connections of mimetic finite difference schemes to the VEM. Introduction to conforming virtual element spaces and the element formulation in 2D and 3D. Numerical implementation of the method and solution of a few benchmark problems will be presented. (Manzini)

3. High-order (k > 1) VEM for the Poisson problem in 2D and 3D and Solution of Biharmonic Equation in 2D: Extension of the VEM to high-order C^0 formulations in 2D and 3D, and a C^1 VEM for the biharmonic equation in 2D. (Manzini)

4. VEM in Solid Mechanics: First-order conforming VEM for isotropic, linear elasticity in 2D and 3D. Stabilization-free virtual element method (SF-VEM) for plane elasticity and extensions of SF-VEM for incompressible linear elasticity will be presented. (Sukumar)

5. VEM in Fluid Mechanics: Formulation of VEM for the Stokes equation (with exact divergence-free elements and weak divergence-free conditions) and time-dependent parabolic problems. (Manzini)

Hands-on Tutorial:

Matlab computer code to solve the 2D Poisson problem using low- and high-order VEM will be made available to participants. Explanation of the code in light of the formulation will be presented, and verification tests of the method (accuracy and rates of convergence) will be assessed through the code. (Sukumar)

Schedule:

Registration: 8:00 am to 8:30 am

Lectures 1-4: 8:30 am to 12:30 am (15 min coffee break after the second lecture)

Lunch: 12:30 pm to 1:30 pm

Lecture 5: 1:30 pm to 2:20 pm

Hands-on Tutorial: 2:30 pm to 4:30 pm

References:

[1] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo, Basic principles of the virtual element method, Math. Models Methods Appl. Sci., 23, 119-214, 2013.

[2] L. Beirao da Veiga, F. Brezzi, L. D. Marini, A. Russo, The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 1541-1573, 2014.

Mrudang Mathur, Stanford University

William Hiesinger, Stanford University

Description:

Augmented reality (AR) is a next-generation visualization paradigm that boasts many advantages over existing data visualizations tools such as images, videos, and scientific visualization software. Specifically, AR visualizations can represent the complete spatiotemporal aspects of data, are interactive in nature, and are easily accessible via smartphones. However, they've found limited adoption in the computational mechanics community to date. This is, in part, due to the domain-specific expertise and proprietary software and hardware previously required to create AR models. To help overcome these challenges, in this short course we will introduce:

1. The fundamentals of computer graphics and 3D modeling required to create augmented reality visualizations.

2. An open-source tool to create, host, and share AR models of scientific results. Specifically, we will help attendees create and share AR models of results from their very own scientific simulations.

Objectives: 

Our objectives for this short course are twofold:

1. to accelerate the adoption of AR visualization within the scientific community and

2. to help researchers improve the accessibility and reach of their scientific results. 

To that end, attendees will leave this course with the requisite knowledge and skills to integrate AR within their own teaching, research, and outreach activities. As a result, they may eschew expert systems and discipline-specific training often needed to visualize and interact with complex spatiotemporal data. This, in turn, may allow a better understanding of data across scientific disciplines and for wider audiences.

Course Materials: 

Course materials will be free to download at: https://github.com/mrudangm-AR/AR_Pipeline

Tentative Schedule:

8:30 am - 9:00 am: Introduction to Augmented/Virtual/Mixed Reality

9:00 am - 10:00 am: Introductory Concepts in Computer Graphics and Rendering

10:00 am - 10:30 am: Coffee Break

10:30 am - 11:10 am: Introduction to Blender

11:10 am - 12:00 pm: Automating Blender with Python

12:00 pm - 1:00 pm: Lunch

1:00 pm - 2:00 pm: Visualizing Lagrangian Analyses

2:00 pm - 3:00 pm: Visualizing Eulerian Analyses

3:00 pm - 3:30 pm: Coffee Break

3:30 pm - 4:10 pm: Hosting and Sharing AR Models

4:10 pm - 4:30 pm: Summary and Future Work