Short Courses

Harry Millwater, The University of Texas at San Antonio

Mauricio Aristizabal, The University of Texas at San Antonio

This hands-on workshop will introduce the use of complex and hypercomplex numbers for automatic differentiation (AD) of numerical and symbolic functions. This method for AD, called HYPAD, has significant advantages, in that

• the methodology is employed “in-place” – that is, the underlying algorithm is merely “augmented” with imaginary degrees of freedom while keeping the fundamental computational implementation largely otherwise unaltered,

• the methodology provides machine precision accuracy, that is, the error in the sensitivities is solely due to the numerical algorithm in which it is employed, and

• once a code is “hypercomplexified”, no further changes are required to assess any numerical derivative, that is, one does not “predefine” the derivative to compute.\

The workshop will cover in detail the use of complex and dual numbers to compute first order sensitivities and hypercomplex bidual numbers to compute 2nd order sensitivities. Extensions to any order sensitivities will also be presented. A working example for a linear elastic finite element analysis will be presented.

The course will be conducted interactively using Python examples and the MultiZ library. Attendees are encouraged to bring a laptop to the course and to download the materials beforehand.

Note - Attendees will obtain a copy of the MultiZ library which can compute arbitrary-order sensitivities for the Python and Fortran languages. Introductory knowledge of Python expected.

HYPAD Workshop Outline

Introduction to HYPAD and course materials

Complex Taylor series expansion (CTSE) method

• CTSE formulation

• Analogy with the finite difference method

• Exercises with simple functions

• Step size study

Numerical integration Application Study

• Simpson’s rule

• Gauss-Legendre quadrature

Newton-Rhapson solver Application Study

ODE solver Application Study

Cauchy-Reimann form

Break

Dual numbers for first order sensitivities

• Definition of dual numbers

• Exercises with simple functions

• Step size study

Numerical integration Application Study

Cauchy-Reimann form

Lunch

Bidual numbers for 2nd order sensitivities

• Basic operations

Exercises with simple functions

• Step size study

Numerical integration Application Study

Newton-Rhapson solver Application Study

Cauchy-Reimann form

Higher order sensitivities

Finite element considerations

Patrick Diehl, Louisiana State University 

Hartmut Kaiser, Louisiana State University 

Steven R. Brandt, Louisiana State University

Many computational engineering codes are based-on the C++ programming language, however, many of these codes do not utilize modern C++ features, especially the features for parallel computations provided by the recent C++ 17 standard. These features can greatly simplify parallel computations using multiple cores.

In this tutorial, participants will learn how to use C++17 functions and objects to write straightforward yet fully parallized code without the need of external tools such as OpenMP. Since the Python programming language is more commonly known by most potential attendees, the team will use Jupyter Notebooks with the Cling extension for C++ to walk attendees step by step through creating a fully parallelized one-dimensional finite element code. Thus, attendees can use a programming environment they are familiar with. Because standard compiler distributions often lack complete support for this feature, our examples will use the HPX framework.

In the second half of the tutorial, the team will demonstrate how users can employ nearly the same syntax to distribute these codes across a cluster. We will use the HPX library to provide the support needed to manage a distributed application.

The example code, the solutions, and the lecture slides will be available on GitHub, so attendees can use them after the short course.

After the short course, attendees will have learned about modern C++ features and how to run a parallel and distributed one-dimensional finite element solver using the mechanics built-in in the C++ 17 and upcoming C++ 20 standard.

• Introduction to C++17 and C++20, Hartmut Kaiser

• Q&A

• Break

• Introduction to asynchronous programming and the parallel algorithms, Patrick Diehl

• Break

• Exercise 1, Parallel implementation of the 2D finite difference scheme

• Break

• Introduction to distributed programming, Steve Brandt

WaiChing Sun, Columbia University

JS Chen, University of California, San Diego

Lab Instructors: Nikolas Vlassis & Bahador Bahmani, Columbia University, Karan Taneja & Kristen Susuki, UC San Diego

This course will be offered to graduate students and researchers to introduce the practical data analytics, dimension reduction, and machine learning techniques, for a variety of science and engineering applications in materials, structures, and systems. This course is designed for the audience with a background in mechanics and/or applied physics. The course will provide an overview of four major categories of machine learning techniques (dimensional reduction of manifold data, geometric learning of graphs, manifold embedding, and deep reinforcement learning) and a data-driven model-free framework. Case studies will be used to demonstrate how these learning techniques have enhanced research and technology advancements. These application problems will include a data-driven model-free paradigm for complex material systems, reduced-order modeling of fracture and thermal fatigue analysis, geometric learning for polycrystal and granular systems, and reinforcement learning-enabled multiscale modeling for decision-making for design-of-experiments. Lecture materials and lab handouts will be provided before the short course.

Target Groups: 

Graduate students, researchers with an understanding of continuum mechanics. A course website will be set up for course materials and sample codes repository before the short course date.

Scientific/Technical areas covered:

1. Manifold learning enhanced data-driven modeling of nonlinear materials

2. Dimension reduction by manifold learning and autoencoders

3. Geometric learning for predicting path-dependent constitutive responses

4. Representation learning for high-dimensional data with physics constraints

5. Reduced-order modeling for fracture and thermal fatigue problems

Provisional schedule

Time Agenda

8 weeks before USNCCM Short course webpage

4 weeks before USNCCM Code repository

Offline lecture Overview (Instructor: Chen & Sun)

Practical tutorial: TensorFlow, PyTorch, Jupiter notebook, cloud computing, code binding (Instructor: Vlassis & Sun)

8:30 am - 9:30 am Deep geometric learning 1: graph embeddings for constitutive modeling and reduced-order modeling with limited data (Instructor: Sun)

9:30 am - 9:45 am Coffee Break

9:45 am - 10:45 am Manifold Based Learning and Data-Driven Computing for Nonlinear Solid Mechanics: Dimension Reduction and Thermodynamics (Instructor: Chen)

10:45 am - 11:00 am Coffee break

11:00 am - noon Jupiter Lab 1

noon - 1:00 pm Lunch break

1:00 pm - 2:00 pm Neural Network Enhanced Computational Mechanics for Localization Problems (Instructor: Chen)

2:00 pm - 2:15 pm Coffee break

2:15 pm - 3:15 pm Deep geometric learning 2: Geometric autoencoder and optimal transport for de-noising of data for elasticity (Instructor: Bahmani, & Sun)

3:15 pm - 3:30 pm Coffee break

3:30 pm - 4:30 pm Jupiter Lab 2

Expected and the minimal number of participants: Expected = 20.

Kevin Linka, Stanford University

Mathias Peirlinck, TU Delft

Ellen Kuhl, Stanford University

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.

SYLLABUS

1.           Introduction to Constitutive Modeling

1.1     History of continuum mechanics

1.2     Kinematics

1.3     Balance equations

1.4     Constitutive equations

2.           Introduction to Neural Network Modeling

2.1     History of neural network modeling

2.2     Neural network input and output

2.3     Network architecture

2.4     Activation functions

3.           Model Discovery for Rubber

3.1     History of constitutive models for rubber

3.2     Constitutive Artificial Neural Networks for rubber

3.3     Rubber in uniaxial tension, equibiaxial tension, and pure shear

3.4     Discovering models for rubber using your own Jupyter notebooks

4.           Model Discovery for the Brain

4.1     History of constitutive models for brain

4.2     Constitutive Artificial Neural Networks for brain

4.3     Gray and white matter tissue in tension, compression, and shear

4.4     Discovering models for brain tissue using your own Jupyter notebooks

5.           Model Discovery for Skin

5.1     History of constitutive models for skin

5.2     Constitutive Artificial Neural Networks for skin

5.3     Skin in uniaxial tension and biaxial extension

5.4     Discovering models for rubber using Jupyter notebooks

6.           The Future of Automated Science

6.1     How can automated model discovery help you?     

6.2     Discussion of perspectives and challenges of automated si

References

[1]  Linka K, Kuhl E. A new family of Constitutive Artificial Neural Networks towards automated model discovery. Comp Meth Appl Mech Eng. 2023; 403:115731. 

[2] St Pierre SR, Linka K, Kuhl E. Automated model discovery for human brain using Constitutive Artificial Neural Networks. 2023. doi:10.1101/2022.11.08.515656.

[3] Linka K, Buganze Tepole A, Holzapfel GA, Kuhl E. Automated model discovery for skin: Discovering the best model, data, and experiment. 2023.

This pre-congress short course will take place on Sunday, July 23, 2023, from 8:30am to 4:30pm. It will cover a theoretical introduction, demos, and hands-on coding activities. You should bring you own laptop and, if you like, your own data. We will provide benchmark data on rubber, brain, and skin, but are equally excited to help you analyze your own experiments. You will get the most out of this course if you familiarize yourself with the references and the code and prepare specific questions, but you are also welcome to attend if you are just curious about the method itself.

Downloads

https://github.com/LivingMatterLab/CANN

OUTLINE

08:30-10:00  Session 01 – Revisiting the Basic Fundamentals 

1. Introduction to Constitutive Modeling

2. Introduction to Neural Network Modeling

10:30-12:00  Session 02 – Getting started: Demos of Model Discovery

3. Model Discovery for Rubber

01:00-02:30  Session 03 – Do it yourself: Discovering your own Models

4. Model Discovery for the Brain

5. Model Discovery for Skin

03:00-04:30  Session 04 – Discussing the Challenges of Model Discovery

          6. The Future of Automated Science

Leszek Demkowicz, The University of Texas at Austin

Stefan Henneking, The University of Texas at Austin

The four-lectures course is addressed to practitioners of standard Finite Element (FE) methods familiar with basic variational formulations, the (Bubnov–)Galerkin method and the standard technology of FEs. The class combines a short introduction to the “Discontinuous Petrov–Galerkin (DPG) Method with Optimal Test Functions” with a crash course on the energy spaces forming the exact sequence and the corresponding conforming FE discretizations. We will introduce the participants to hp3D—a 3D MPI/OpenMP code supporting hp-discretizations of the exact-sequence elements on hybrid (tets, cubes, prisms, pyramids) meshes and demonstrate how to implement the DPG method in such a framework. On the application side, we will focus on wave propagation problems: time-harmonic acoustics, Maxwell’s equations, and elastodynamics.

Syllabus:

1. 1. 1. Lecture 1

         1. • Examples of variational formulations with symmetric and non-symmetric functional setting; brief introduction to energy spaces.

            • A crash course on H1, H(curl), H(div), and L2-conforming finite elements.

      2. Lecture 2

         1. • Introduction to the hp3D code.

            • Examples of applications of the Bubnov–Galerkin method.

      3. Lecture 3

         1. • A crash course on the DPG method.

            • Implementation of DPG in the hp3D code.

      4. Lecture 4

         1. • MPI/OpenMP parallel computation with the hp3D code.

            • Examples of applications of the DPG method.

Course Material:

1. L. Demkowicz. Lecture Notes on Energy Spaces. Technical Report 13, ICES, 2018. https://www.oden.utexas.edu/media/reports/2018/1813.pdf

2. L. Demkowicz. Lecture Notes on Mathematical Theory of Finite Elements. Technical Report 11, Oden Institute, June 2020. https://www.oden.utexas.edu/media/reports/2020/2011.pdf

3. S. Henneking and L. Demkowicz. Computing with hp Finite Elements III. Parallel hpCode. 2022. In preparation, available upon request.

Software:

• hp3D open-source library, GitHub: https://github.com/Oden-EAG/hp3d

• hp3D user manual, arXiv:2207.12211 (2022): https://arxiv.org/pdf/2207.12211.pdf

Pablo Seleson, Oak Ridge National Laboratory

John Foster, The University of Texas at Austin

David Littlewood, Sandia National Laboratories

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 - 13:00 Lunch

13:00 - 13:30 Modeling Failure and Damage

13:30 - 14:00 Multiphysics Modeling in Peridynamics

14:00 - 14:30 Multiscale Modeling in Peridynamics

14:30 - 14:45 Break

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

15:30 - 16:00 Practice Session

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

Romit Maulik, Argonne National Laboratory

We will provide a tutorial and hands-on session for integrating linear algebra and machine learning libraries in Python with OpenFOAM - a popular open-source CFD tool. The course will have a lecture+tutorial session to introduce participants with the basics of coupling C++ and Python codes. Subsequently, there will be a hands-on session where participants will be assisted with installing, compiling, and running the developed software (available here: https://github.com/argonne-lcf/PythonFOAM). Video lectures for the tutorial are already available at: https://www.youtube.com/watch?v=-Sa2OEssru8. More content will be added for this short course.

1. Introduction - 30 minutes

Introducing instructors and surveying audience for expectations from session. Quick questions to establish OpenFOAM proficiency and domains of research. 

2. Literature survey – 30 minutes

Review of certain popular machine learning algorithms that have successfully used with CFD problems. Focus on results from PythonFOAM.

3. Minimum working example – 60 minutes.

Coupling demo on minimum working example with C++ and Python for a 1D problem. Hands-on to help attendees compile and run demo.

4. Installing PythonFOAM, building new libraries, applications with Python compatibility - 120 minutes

Tutorial of Docker to help install PythonFOAM on *nix machines. Hands-on walk through of various examples in PythonFOAM (such as for in-situ autoencoders, in-situ singular value decompositions). 

5. Hackathon for OpenFOAM users to deploy their own projects in PythonFOAM for surrogate modeling – 120 minutes.

Users are encouraged to interface their CFD projects with PythonFOAM capabilities and/or extend PythonFOAM capabilities.

Martin van der Schelling, Delft University of Technology

Miguel Bessa, Brown University

Jiaxiang Yi, Delft University of Technology

Summary:

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.

3. Machine learning, where we either train a surrogate model to fit our experimental findings.

4. Optimization, where we iteratively improve the model to get an optimum 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 [2, 3] and, more importantly, be able to use different tools to perform new data-driven investigations to pursue your own research endeavor. 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:

Duration Description

1h00 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.

10 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.

10 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.

10 min Break

1h30 Part 4: Practical session –  constitutive law prediction for composites

• Case study: use the F3DASM framework to generate microstructures, do finite-element simulation and establish a machine learning model for constitutive law prediction [3].

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] Bessa, M. A., Bostanabad, R., Liu, Z., Hu, A., Apley, D. W., Brinson, C., Chen, W., & Liu, W. K. (2017). A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality. Computer Methods in Applied Mechanics and Engineering, 320(April), 633–667. https://doi.org/10.1016/j.cma.2017.03.037

[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

[3] Dekhovich, A., Turan, O. T., Yi, J., & Bessa, M. A. (2022). Cooperative data-driven modeling. 1–13. http://arxiv.org/abs/2211.12971

Joshua Robbins, Sandia National Laboratories

Marshall Galbraith, Massachusetts Institute of Technology

John Dannenhoffer III, Syracuse University

Brett Clark, Sandia National Laboratories

This course will provide a hands-on introduction to the open-source Plato and Engineering Sketch Pad (ESP) software packages for optimization-based engineering design. Students will be introduced to the mathematics of gradient-based shape optimization as well as the key features and design of Plato and ESP.  This course will provide a foundational knowledge of design optimization and the Plato and ESP software that researchers and practitioners can build upon for algorithm development and/or real-world engineering applications.

Schedule:

8:30-9:00     (Optional) Get help with ESP installation.

9:00-9:30 Introduction to shape optimization

9:30-12:00 Introduction to the Engineering Sketch Pad

Goal: Students will learn the basic functionality of ESP.

Outcome:  Students are able to create parameterized geometry from scratch and export an attributed mesh for subsequent shape optimization.

Note:  This session includes a 15-minute break.

12:00-1:00 (Optional) Working lunch: Instructors will be available to students for questions and one-on-one help.

1:00-3:30     Introduction to Plato

Goal: Students will learn the basics of how to use Plato for shape optimization.

Outcome: Students know how to set up and run optimization problems using meshes from the morning session.

Note:  This session includes a 15-minute break.

Important:  Students wishing to participate in the hands-on ESP tutorial will need to bring a laptop computer with ESP installed. Download the latest version from acdl.mit.edu/ESP/PreBuilts.

C. Alberto Figueroa, University of Michigan

Abhilash Malipeddi, University of Michigan

Liz Livingston, University of Michigan

The goal of this hands-on software Short Course is to expose participants to a variety of novel functionalities for advanced simulation of cardiovascular multi-physics, using the open-source framework CRIMSON (www.crimson.software). The anticipated target audience of this Short Course is any researcher interested in multi-physics cardiovascular modeling and transport areas.

The Short Course will consist of four learning modules, see table below.

Module I: In the first module, we will provide a demonstration of CRIMSON’s GUI, including how to process medical image data, import geometric models created with CAD software, perform mesh generation, specify boundary conditions, and run simulations.

Module II: Dynamically-adapting inflow and outflow boundary conditions to simulate cardiovascular regulation and control.

Module III: Simulation of transport of multiple scalar species coupled with flow.

Module IV: Particle-laden flow simulations using an Euler-Lagrange approach on unstructured grids, where particle-fluid and particle-particle interactions are included.

Important links:

• Flow Solver repo: https://github.com/carthurs/CRIMSONFlowsolver

• GUI repo: https://github.com/carthurs/CRIMSONGUI

• PLOS Computational Biology paper: https://doi.org/10.1371/journal.pcbi.1008881  

Oscar Lopez-Pamies, University of Illinois Urbana-Champaign

Aditya Kumar, Georgia Institute of Technology

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 the end of the 1990s) by the mathematics and computational mechanics communities aimed at describing when and how fracture nucleates and propagates in solids under arbitrary mechanical loads in a computationally tractable manner. These efforts comprise three pivotal ideas: (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 [3-5]. The latter two ideas constitute the phase-field approach to fracture.

Specifically, the course will focus on the phase-field approach to fracture in 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 FEniCS library [6]) will be given to solve representative 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 Python 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, Ravi-Chandar K, Lopez-Pamies O (2022) Int J Fract. 237, 83–100.

6. FEniCS computing platform, https://fenicsproject.org/.

Syllabus:

Hours 1-3: The Theory

• Griffith idea for fracture

• Fracture propagation as a variational problem

• Regularization of Griffith-type surface energy and introduction to phase-field

• Euler-Lagrange equations of the variational problem

• Ingredients for describing fracture nucleation at large

• Concept of strength surface

• Generalizing the Euler-Lagrange equations of the variational problem to account for the strength surface

Hours 4-6: The Numerical Implementation

• Weak form and finite element formulation of the PDEs

• Staggered formulation for solving coupled PDEs

• Choice of regularization length scale

• Calibration of toughness and strength parameters

• Representative boundary-value problems:

  • ​Surfing problem for fracture propagation

  • Indentation problem with a cylindrical indenter

  • Nucleation from a V-notch

  • Mixed-mode propagation in a compact tension test

  • Nucleation and propagation in an elastomeric poker-chip specimen

Miguel Bessa, Brown University

Bernardo Ferreira, Brown University

Computer simulations dramatically evolved to become predictive but they remain time-consuming. This short-course presents a new Python open-source code designed to accelerate material simulations under nonlinear and irreversible deformation: CRATE.

Clustering-based Nonlinear Analysis of Materials (CRATE) performs accurate and efficient multi-scale nonlinear analyses of solid materials by leveraging first-order computational homogenization and clustering-based reduced-order models. Provided a given microstructure, it admits general nonlinear constitutive models, allows the enforcement of general non-monotonic macro-scale loading paths, and delivers several output files for a suitable post-processing. Particular focus is given to a clustering-based reduced-order model called Self-consistent Clustering Analysis (SCA) [1], which is adopted to solve the micro-scale equilibrium problem. A recently proposed extension coined Adaptive Self-consistent Clustering Analysis (ASCA) [2] is also covered, where clustering adaptivity is leveraged to handle localization and damage phenomena.

The learning objectives of this course are the following:

• Determining mechanically-informed material clusters with unsupervised machine learning (offline stage);

• Fast prediction of material's response (online stage);

• Learning the code structure from characterization of microstructure, property prediction to post-processing of results;

• Learning how to extend CRATE with user-defined features (e.g., constitutive models and clustering algorithms) for solving new problems.

[1] Liu, Z., Bessa, M., and Liu, W. K. (2016). Self-consistent clustering analysis: An efficient multi- scale scheme for inelastic heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 306:319–341.

[2] Ferreira, B.P., Andrade Pires, F.M. and Bessa, M.A., (2022). Adaptivity for clustering-based reduced-order modeling of localized history-dependent phenomena. Computer Methods in Applied Mechanics and Engineering, 393.

Program:

Duration Description

1h00 Part 1: SCA fundamentals

Brief introduction to materials multi-scale simulation based on computational homogenization. Overview of SCA method. Description of offline-stage (learning) and online-stage (prediction) fundamental concepts. Introduction to CRATE.

10 min Break

1h30 Part 2: SCA's offline-stage

Description of CRATE´s API and code structure. Hands-on examples on fast linear elastic simulations, clustering features and algorithms, and computation of clustering interaction tensors.

10 min Break

1h30 Part 3: SCA's online-stage

Description of CRATE´s API and code structure. Hands-on examples on integration of constitutive models, enforcement of general loading paths, solution of Lippmann-Schwinger equilibrium equations, and post-processing of results.

10 min Break

1h30 Part 4: ASCA fundamentals and showcases

Overview of ASCA clustering adaptivity procedures. Description of CRATE´s API and code structure. Illustrative examples of clustering adaptivity in action when handling plastic strain localization.

Closing remarks

Manuel Rausch, The University of Texas at Austin

Mrudang Mathur, The University of Texas at Austin

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. 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.

Course Objectives:

Our objectives for this short course are twofold: (i) to accelerate the adoption of AR visualization within the scientific community and (ii) 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/SoftTissueBiomechanicsLab/AR_Pipeline

Tentative Schedule:

Time Topic

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

N. Sukumar, University of California, Davis

Gianmarco Manzini, Los Alamos National Laboratory

In this short course, we will present the foundations and applications of the Virtual Element Method (VEM) [1] 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. 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 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 first-order and serendipity (k = 2,3) VEM for plane elasticity will be presented. (Sukumar)

5. VEM in Fluid Mechanics: Formulation of VEM for the Stokes equation (exact discrete divergence-free element) in 2D and its extension for the time-independent Navier-Stokes equations in 2D. (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. (Manzini and 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

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

Wing Kam Liu, Northwestern University, Co-Founder of HIDENN-AI, LLC

Dong Qian, University of Texas at Dallas and Co-Founder of HIDENN-AI, LLC

In engineering and physical science, we routinely deal with discrete data. Depending on how much we know about the system (the system behavior may be completely, partially, or hardly understood), these data need to be curated and analyzed to obtain mechanistic insight. Numerical methods such as finite difference, finite elements, or similar tools have been widely used to deal with such discrete data[i]. However, dealing with high-resolution data (e.g., image or mesh data) for extreme-scale engineering problems is computationally challenging with traditional numerical methods. In recent times, researchers have resorted to data science techniques, especially deep learning to use the data for system characterization and analysis. However, little mechanistic insight can be obtained from pure data-driven approachesi. Hierarchical Deep Learning Neural Network (HiDeNN) and Deep-Learning Discrete Calculus (DLDC)[ii] research aim to bridge numerical techniques, mechanistic knowledge, and data science methods and propose computationally efficient tools designed for extreme-scale science and engineering problems. DLDC is also developed as a series of lectures for STEM education and frontier research, as it offers a new perspective on numerical methods and deep learning.

The term “deep learning” refers to a subset of artificial intelligence (AI) that uses universal approximation theory through a set of neurons, activation functions, and optimizable hyperparameters to approximate any linear or non-linear relationships. Calculus is a branch of mathematical science that deals with changes in a system. DLDC[iii] is a computational method that uses deep learning constructs to mimic numerical methods based on discrete calculus. The DLDC uses elements of data science (data generation and collection, feature extraction, dimension reduction, reduced order models, regression and classification, and system and design) to solve science and engineering problems[iv]. The DLDC method offers enhanced computational efficiency and accuracy compared to traditional numerical methods, especially when dealing with high-resolution experimental/computational datasets.

Let us consider designing a drone body frame with extremely high resolution (e.g., 10 billion degrees of freedom, DoFs). Most of the computing resources are devoted to solving the physical response of the system, which is excessively demanding. To solve such problems, the DLDC is extended to Hierarchical Deep Learning Neural Network (HiDeNN) and convolution hierarchical deep-learning neural network (C-HiDeNN)[v]and its reduced order method, i.e., C-HiDeNN tensor decomposition (C-HiDeNN-TD)[vi], that enables highly accurate solutions with less computational overhead. Convolution is a mathematical operation well-known in applications such as signal processing[vii] and convolutional neural networks (CNN)[viii]. C-HiDeNN builds adaptive (the convolution filter can have varying sizes and values like CNN) convolution filters that cover a domain outside of an element (a.k.a. patch domain) so that we can interpolate a larger domain with higher smoothness and completeness without increasing the global degrees of freedom like higher-order FEM. C-HiDeNN-TD[ix] is a reduced-order version of C-HiDeNN where we only use 1-dimensional convolutional filters. Based on the concept of separation of variables, the Tensor Decomposition (TD) method converts an “n”-dimensional problem into “n” 1-dimensional problems for some number of decomposition modes. Since solving a full-dimensional matrix equation takes most of the computation time for extremely large-scale problems, dividing the equation into multiple small equations can greatly reduce the computational burden. Orders of magnitude speedup and improvement in the accuracy of the solution can be obtained by C-HiDeNN-TD. Another extension of C-HiDeNN might be the convolution isogeometric analysis (C-IGA) that can recover exact geometry while maintaining higher order continuity and retaining the Kronecker delta property of the underlying shape functions.

This short course will introduce and demonstrate how to apply a) HiDeNN and C-HiDeNN, b) C-IGA, and c) C-HiDeNN-TD. In the later part of the course, Graphical Processing Unit (GPU) acceleration of DLDC[x] will be demonstrated via Google Colab with the JAX[xi] library in Python. Participants are welcome to bring their laptops, and there is no need for installation/registration during this demonstration session. The application examples will focus on using the DLDC technique for topology optimization and multiscale materials design. After finishing the course, the attendees will be able to understand and apply the DLDC methods for solving engineering problems that require very accurate solutions given high-resolution data.

Outline of the Course:

Timeline Lecture Content

8:30 - 10:00 History of FEM and Mechanistic data science that leads to HiDeNN History of FEM

Basics of FEM (weak form, shape function)

From FEM to meshfree/IGA

Emergence of MDS and machine learning

Basics of DNN and machine learning, programming in Python

10:00 - 10:30 Coffee break

10:30 - 12:00 Background: Hierarchical Deep-learning Neural Network (HiDeNN) Three building blocks of HiDeNN

HiDeNN-Finite Element Method (HiDeNN-FEM)

R-adaptivity

Introduction to tensor decomposition

HiDeNN-tensor decomposition

12:00 - 1:00 Lunch

1:00 - 2:15 Extension of HiDeNN-FEM to meshfree, enrichment and IGA approximations HiDeNN-meshfree approximation

HiDeNN-enrichments

HiDeNN-IGA

Examples

2:15 - 3:15 Convolutional Hierarchical Deep-learning Neural Network (C-HiDeNN) Convolution patch functions

Graph theory for nodal connectivity

Demonstration of C-HiDeNN

Preliminary results of C-HiDeNN

3:15 - 3:45 Coffee break

3:45 - 4:30 HiDeNN-FEM for nonlinear problems Basics of nonlinear FEM (total Lagragian formulation)

Building blocks for nonlinear HiDeNN-FEM

Linearization and Newton’s method

Application to nonlinear elasticity and plasticity

4:30 - 5:30 Applications C-HiDeNN-TD for Topology Optimization

C-HiDeNN for Multiscale Materials Design

[i] W.K. Liu, S. Li, H.S. Park, Eighty Years of the Finite Element Method: Birth, Evolution, and Future, Archives of Computational Methods in Engineering pp.1–23 (2022).

[ii] Saha S, Park C, Knapik S, Guo J, Huang O and Liu WK (2023), Deep Learning Discrete Calculus (DLDC): A Family of Discrete Numerical Methods by Universal Approximation for STEM Education to Frontier Research. Computational Mechanics.

[iii] Saha S, Park C, Knapik S, Guo J, Huang O and Liu WK (2023), Deep Learning Discrete Calculus (DLDC): A Family of Discrete Numerical Methods by Universal Approximation for STEM Education to Frontier Research. Computational Mechanics.

[iv] Liu, Gan, Fleming, “Mechanistic Data Science for STEM Education and Applications,” Springer, 2021

[v] Lu Y, Li H, Zhang L, Park C, Mojumder S, Knapik S, Sang Z, Tang S, Wagner G and Liu WK (2023) Convolution Hierarchical Deep-learning Neural Networks (C-HiDeNN): Finite Elements, Isogeometric Analysis, Tensor Decomposition, and Beyond. Computational Mechanics.

[vi] Li H, Knapik S, Li Y, Guo J, Lu Y, Apley DW and Liu WK (2023) Convolution-Hierarchical Deep Learning Neural Network-Tensor Decomposition (C-HiDeNN-TD) for high resolution topology optimization. Computational Mechanics.

[vii] Liu WK, Jun S and Zhang YF (1995) Reproducing kernel particle methods. International journal for numerical methods in fluids 20: 1081-1106 DOI. Liu WK, Han W, Lu H, Li S and Cao J (2004) Reproducing kernel element method. Part I: Theoretical formulation. Computer Methods in Applied Mechanics and Engineering 193: 933-951 DOI.

[viii] LeCun, Y., & Bengio, Y. (1995). Convolutional networks for images, speech, and time series. The handbook of brain theory and neural networks, 3361(10), 1995.

[ix] Zhang, L., Lu, Y., Tang, S., & Liu, W. K. (2022). HiDeNN-TD: Reduced-order hierarchical deep learning neural networks. Computer Methods in Applied Mechanics and Engineering, 389, 114414.

[x] C. Park, Y. Lu, S. Saha, T. Xue, J. Guo, S. Mojumder, G. Wagner, W. Liu, Convolution Hierarchical Deep-learning Neural Network (C-HiDeNN) with Graphics Processing Unit (GPU) Acceleration, Computational Mechanics (2023).

[xi] https://jax.readthedocs.io/en/latest/

Gianluca Geraci, Sandia National Laboratories

Alex Gorodetsky, University of Michigan

John Jakeman, Sandia National Laboratories

This short course will provide an introduction to multi-fidelity methods for uncertainty quantification that can be used to optimally balance the allocation of computational resources among models with varying accuracy and cost. Multi-fidelity methods combine a limited number of high-fidelity model runs, used to maintain predictive accuracy of complex fine scale phenomena, with a larger number of evaluations of (possibly multiple) lower fidelity models, which allow greater exploration of the model uncertainty space. The course will begin with a introduction to UQ using a single high-fidelity model, then incrementally introduce multi-fidelity methods of increasing complexity that address the various challenges of single-fidelity UQ. We will cover important theoretical and practical aspects of each method introduced. The main focus of the course will be on sampling methods, i.e. methods that can be derived from Monte Carlo. Nevertheless, a brief introduction to surrogate-based MF approaches will also be included to provide the students with a basics understanding of the literature. At the end of the short course, we hope each student will walk away with the understanding and access to tools to immediately begin applying multi-fidelity methods to their problem of interest.

Course Audience

Graduate students, postdocs, computational scientists interested in applying uncertainty quantification for non-trivial computational problems and applications 

Prerequisites

Knowledge of uncertainty quantification is not required

Software

Solution to exercises will be provided in Python (participants could use any software during the numerical sessions, e.g., Matlab, R, etc.), however the Python-based software PyApprox (https://sandialabs.github.io/pyapprox/index.html) will be used for most of the tutorials. The participants are encouraged to familiarize with it, but this is not required to participate – The solutions to all problems will be discussed and the participants will be able to complete the exercises at their own pace, after the course.

Course Material

Material will be provided to the participants in advance. Tutorials will be supported by simple Python scripts. PyApprox tutorials will be provided during the course.

Course Outline

Session 1 (2 hrs)

• Opening remarks, logistics

• Introduction to Uncertainty Quantification

• Brief overview of forward UQ approaches

• Sampling-based approaches: Monte Carlo and control variate

• Numerical demonstrations (30 min): sampling-based approaches in action

Break (45 min)

Session 2 (2 hrs)

• From one model to multiple low-fidelity models: how can we apply the concept of control variate to realistic scientific and engineering models? This session will cover several estimators from Multilevel Monte Carlo to Multi-fidelity Monte Carlo and Approximate Control Variate

• Numerical demonstrations (30 min): MF sampling-based approaches in action

Break (45 min)

Session 3 (2 hrs)

• Brief intro on MF surrogate based approaches (1 hr)

• Advanced topics on MF UQ: challenges, open questions and recent trends (30 min)

• Numerical demonstrations (30 min): MF surrogate-based approaches in action

Pamela Abshire, University of Maryland

Jennifer Blain Christen, Arizona State University

Nicole McFarlane, University of Tennessee

Maira Samary, Boston College

Stephen D. Senturia, Massachusetts Institute of Technology, Emeritus

Virtual attendance now available – please register here https://www.theacademiclife.org/events.

The Academic Life Faculty-Development Workshops are designed for those either already in or seeking STEM-oriented careers in academia. Our goal is to explore aspects of academic life that can be shrouded in privacy and secrecy, and through discussion of these issues, provide meaningful mentoring to aspiring academics.

The central theme of our workshops is that dramatizations can illustrate and highlight the challenges aspiring academics may face. Each unit includes extensive opportunities for discussion and sharing of personal experiences.

While these workshops grew out of an initial focus on gender issues, the broader set of topics now covered provide guidance to all who have a successful academic career as their goal.

NSF-sponsored Scholarship Aid is now available to cover the cost for the event. Click here to apply.

“The Academic Life” Faculty-Development Workshops are supported in part by the National Science Foundation under NSF Grant 1844528.

Program Details

Tenure: We present the dramatization Anatomy of a Tenure Case, based on a realistic but fictional tenure decision at a high-tech university. We follow the candidate and her mentor through the decision year, ending at the point at which the department must make a decision. We vote as a group, and discuss our votes.

Break

Publications: Professor Senturia presents a lecture entitled “Why (and How to) Get Published: Wisdom from a Former Journal Editor.” It is based on his 36 years on the MIT Faculty, including 17 years as an editor, and discusses publication strategies, structuring good papers, and responding to peer reviews.

LUNCH

Peer Review: We present the dramatization Power and Plagiarism, in which a junior faculty person is asked to review a paper submitted by a very senior person, and there are questions about the originality of the submitted work.

Microaggressions: The term “microaggressions” refers to behaviors, intended or not, that are hurtful to individuals. Our dramatization, On the Receiving End, features a female junior faculty member who talks to her mother and then to friend about what could be called a really bad day at work.

Break

Imposter Syndrome: Every professional has at one time or another worried about whether he or she is qualified for their own position. That self-doubt is often called “The Imposter Syndrome.” Our dramatization Am I an Impostor is based on a real incident from one of our presenters, and offers some guidance on how to deal with such situations.

Closing: Participants are asked to fill out an anonymous survey critiquing the workshop, and, as a thank-you, will receive a signed and inscribed copy of One Man’s Purpose, the novel by Professor Senturia on which several of our dramatizations are based.