Schedule for: 25w5441 - Geometric Mechanics Formulations for Continuum Mechanics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk, I will survey Lagrangian, Hamiltonian, and Dirac mechanics formulations of mechanics, their associated geometric structures, and Noether conservation laws. I will also explore the issue of symmetry and how it leads to reduced variational principles and reduced geometric structures. I will then discuss the generalization to field theories using the multisymplectic formulation, and the challenges associated with gauge symmetries. I will then describe geometric structure-preserving discretizations using group-equivariant interpolation spaces and multisymplectic variational integrators before describing the connection between cochain projections such as finite element exterior calculus and variational discretizations of Lagrangian field theories.

Short introductions of 30s to 1min from all participants.

Space-time discretization methods are becoming increasingly popular, since they allow parallelization and adaptivity in space and time simultaneously. However, in order to exploit these advantages, one needs to have a complete numerical analysis of the corresponding Galerkin methods. In the case of time-dependent Maxwell's equations, this motivates us to consider the de Rham complex in 4d. Although the analysis can be done with little pain with FEEC, implementation requires that we have a closer look at the corresponding vector proxies using Riemannian metric. In this talk, I want to briefly summarize how these look like in order to open the discussion to the modelling choices that may be useful for discretization.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Differential complexes encode important algebraic and differential structures of physics models. Different problems involve different differential structures and complexes. For grad-div-curl related problems, such as those from electromagnetism and fluid dynamics, the de Rham complex plays a fundamental role. For other problems, such as those from continuum mechanics, differential geometry, and general relativity, other complexes are required, such as the so-called elasticity (Kröner, Calabi) complex. These complexes and their properties can be systematically derived from the de Rham complex via a Bernstein–Gelfand–Gelfand (BGG) construction. There appears to be a neat correspondence between a large class of continuum mechanics models and the BGG machinery. Hence, differential complexes also provide a new angle for developing mechanics models and shed light on their structure-aware formulation. In this talk, we discuss the BGG machinery and their correspondence to elasticity, microstructures (micropolar models), continuum defects, dimension reduction, and multi-dimensional models. This paves a way for structure-preserving discretization.

Simulating fluid dynamics on curved surfaces is crucial for applications in computer graphics and biological systems. The vorticity-streamfunction formulation is particularly appealing due to its scalar representation, which avoids the complexities of covariant differentiation of velocity-based approaches. However, accurately modeling viscous vortices on surfaces, especially in the presence of general topology and curvature, remains underexplored. We derive the vorticity form of the Navier–Stokes equation for general surfaces, explicitly addressing the often-overlooked influence of Gaussian curvature on viscosity. This curvature-dependent term plays a key role in ensuring proper conservation properties in isometric flows and governing energy transfer between the cohomological component and viscosity. Another critical aspect of vorticity-based methods is prescribing viscous boundary conditions. We analyze these conditions in the most general setting, focusing on the free-slip condition at curved boundaries. Surprisingly, this condition can be formulated as a viscous response to the delta-concentrated Gaussian curvature at the boundary, offering a perspective that leads to a stable numerical implementation. We also speculate on the curvature dependence of the free-slip condition in explaining the well-known Kutta condition. Finally, we extend fluid simulations to non-orientable surfaces, such as the Möbius band, and present facts about different surface Laplacians and their kernels, including harmonic vector fields and Killing fields.

Tensor products of differential forms play a prominent role in certain differential complexes like the elasticity complex, the Hessian complex, and the div-div complex. We construct piecewise constant finite element spaces for such tensors. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for algebraic curvature tensors.

In order to preserve structure in a numerical algorithm, one needs to identify what is meant by structure both before and after discretization. In this talk I will review various kinds of structure, including, e.g., conservative, Hamiltonian, and dissipative. Symplectic, Poisson, and conservative integrators, for both finite and infinite-dimensional systems will be discussed. Dissipative systems that are thermodynamically consistent are also of interest. The metriplectic 4-bracket algorithm for identifying and creating such systems and how this structure can naturally lead to thermodynamically consistent discretizations will be discussed. Below are some recent relevant papers. P. J. Morrison and M. Updike, "Inclusive Curvature-Like Framework for Describing Dissipation: Metriplectic 4-Bracket Dynamics," Physical Review E 109, 045202 (22pp) (2024). A Zaidni, PJ Morrison, and S Benjelloun, "Thermodynamically Consistent Cahn-Hilliard-Navier- Stokes Equations using the Metriplectic Dynamics Formalism." Physica D 468, 134303 (11pp) (2024). W. Barham, P. J. Morrison, and A. Zaidni, "A Thermodynamically Consistent Discretization of 1D Thermal-Fluid Models Using their Metriplectic 4-Bracket Structure," arXiv:2410.11045v1 [physics.comp-ph] 14 Oct. 2024. Communications in Nonlinear Science and Numerical Simulations, to appear. A. Zaidni and P. J. Morrison, "Metriplectic 4-Bracket Algorithm for Constructing Thermodynamically Consistent Dynamical Systems," arXiv:2501.00159v1 [physics.flu-dyn] 30 Dec 2024.

I will review recent advances in Variational Thermodynamics, an extension of the critical action principle of mechanics that incorporates irreversible processes such as viscosity, heat and matter exchange, and chemical reactions. I will discuss applications to thermodynamically consistent modeling and structure-preserving discretization, with examples from fluid and plasma physics.

Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications to optimization and control, we present a new Type II variational approach for Hamiltonian systems, based on a virtual work principle that enforces Type II boundary conditions through a combination of essential and natural boundary conditions. We will discuss how this variational principle can be formulated on vector spaces and subsequently extend it to parallelizable manifolds, intrinsic manifolds, and infinite-dimensional Banach spaces. This is joint work with Melvin Leok.

Weather and ocean prediction models involve a combination of physical modelling and observational data. To develop models in which the data does not alter the underlying structures, several choices are available. In this talk, I discuss stochastic approaches to formulating models in continuum mechanics that have degrees of freedom which can be parametrised by data.

We consider neural networks (NN) as discretizations of continuous dynamical systems. There are two relevant systems: the NN architecture on one side and the gradient flow for optimizing the parameters on the other. In both cases, stability properties of the discretization methods can be relevant e.g. for adversarial robustness. Moreover, to prevent the problem of exploding or vanishing gradients, it is common to consider NNs whose feature space and/or parameter space is a Riemannian manifold. We investigate the stability of the explicit Euler method defined on Riemannian manifolds, namely the Geodesic Explicit Euler (GEE). We provide a general sufficient condition which ensures stability in any Riemannian manifold. Whenever the manifold has constant sectional curvature, such condition can be turned into a rule for choosing the stepsize.

Open discussion on the topic of Geometric Mechanics. - Discuss content of presentations - Relevant future developments - Challenges - etc.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Introduction of Mentors and Mentees.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This lecture gives a sneak peek into the world of structure-preserving discretisation methods with a focus on partial differential equations from fluid dynamics and plasma physics. After a short motivation of the usefulness of preserving structure in numerical algorithms, some exemplary techniques will be introduced, specifically discrete differential forms and the discretisation of Poisson and metriplectic brackets using particle and Galerkin methods.

I present a new algorithm for the finite element approximation of the spectral problem of the curl operator that exploits a tree-cotree decomposition of the graph relating the degrees of freedom of the Lagrangian finite elements and those of the Nédélec finite elements. It reduces significantly the dimension of the algebraic eigenvalue problem to be solved and it is well adapted to domains of general topology and to high order finite element approximations. Numerical experiments in the lowest order case are included in order to illustrate the performance of the method.

A key ingredient in the formulation of stable numerical methods for fluid flows and electromagnetism is the construction of a discrete de Rham complex. In this talk, we'll discuss the construction of such a complex using hierarchical B-splines (HB-splines). HB-splines generalize tensor-product B-splines to allow for local refinement and are well-suited for the construction of high-order adaptive finite element methods. Given a hierarchical mesh of an $n$-dimensional hypercube, we will present local and sufficient conditions that ensure that the corresponding HB-spline de Rham complex is exact (10.1007/s10208-024-09659-6). We will show examples that satisfy our conditions as well as those that violate them. We will also relate our results to those of Evans et al. (10.1093/imanum/dry077) and highlight the strength of their joint implications.

Studying the de Rham complex is a natural choice when working with problems in electromagnetics and fluid mechanics. By discretizing the complex correctly, it is possible to attain stable numerical methods to tackle these problems. An important consideration when constructing the discrete complex is that it must preserve the cohomology structure of the original one. This property is not guaranteed when the discrete function spaces chosen are hierarchical B-splines. Research shows that a poor choice of refinement domains may give rise to spurious harmonic forms that ruin the accuracy of solutions, even for the simplest partial differential equations. Another crucial aspect to consider in the hierarchical setting is the notion of admissibility, as it is possible to obtain optimal convergence rates of numerical solutions by limiting the multi-level interaction of basis functions. We will focus on the two-dimensional de Rham complex over the unit square $\Omega \subseteq \mathbb{R}^2$. In this scenario, the discrete de Rham complex should be exact, and we provide both the theoretical and the algorithm-implementation framework to ensure this is the case. Moreover, we show that, under a common restriction, the admissibility class of the first space of the discrete complex persists throughout the remaining spaces. Finally, we include numerical results that motivate the importance of the previous concerns for the vector Laplace and Maxwell eigenvalue problems.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Our formulation of vector bundle valued DEC is for the category of ordered simplicial complexes with morphisms being order-preserving abstract simplicial maps. For simplicial complexes, simplicial maps play the role that smooth maps play for manifolds. The naturality properties in the discrete world, such as d-nabla and cup product commuting with pullbacks, are with respect to simplicial maps. Thus naturality for the ordered case requires a mechanism to generate order-preserving maps. We use the subdivision functor to convert ordinary simplicial complexes and ordinary simplicial maps into ordered versions. This allows us to avoid having to give a total ordering on the vertices. This also allows us to relate our framework to discrete formulations that associate fibers with simplices of all dimensions. It also paves the way for future work on incorporating the metric of base manifolds the way it is done in scalar valued DEC using primal and dual complexes. Joint work with Daniel Berwick-Evans and Mark Schubel.

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions.

Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterized by a conservative part, which can be modeled as a Hamiltonian term, and a nonconservative part, in the form of gradient flow dissipation. Traditional numerical approximations of this class of problem typically fail to retain the separation into conservative and nonconservative parts hence leading to unphysical solutions. In this talk we present a mixed variational method that gives a semi-discrete problem with the same geometric structure as the infinite-dimensional problem. As a consequence the conservation laws and the dissipative terms are retained. This is joint work with Damiano Lombardi, Inria Paris (France).

Smoothed particle hydrodynamics (SPH) is a Lagrangian numerical method that approximately solves the Eulerian equations of fluid dynamics. Moreover, it has a Hamiltonian structure, as well as the Euler equations of fluid dynamics. Here, we extend SPH to approximate the Symmetric Hyperbolic Thermodynamically Consistent (SHTC) equations, which model dynamics of both complex fluids and solids at once. The Hamiltonian structure of SHTC is then reflected in the structure of the numerical scheme (SHTC-SPH).

Geometric particle-in-cell discretizations have been derived based on a discretization of the fields that is conforming with the de Rham structure of the Maxwell’s equation and a stan- dard particle-in-cell ansatz for the fields by deriving the equations of motion from a discrete action principle. While earlier work has focused on finite element discretization of the fields based on the theory of Finite Element Exterior Calculus, we propose an alternative formulation of the field equations that is based on the ideas conveyed by mimetic finite differences. The needed duality being expressed by the use of staggered grids. We construct a finite difference formulation based on degrees of freedom defined as point values, edge, face and volume integrals on a primal and its dual grid. In numerical experiments, we verify the conservation properties of the novel method and study the influence of the various parameters in the discretization.

Following the work by Francois Dumoures and Francois Gay-Balmaz, J. Comp. Dyn., 2025, a space-time spectral element formulation for barotropic, inviscid flow will be presented. The resulting flow satisfies conservation of mass (trivially), conservation of linear momentum, conservation of angular momentum and conservation of total energy. The Mach number is determined by material constants and initial and boundary conditions. Low Mach number flows -- which generally pose problems in Eulerian formulations -- do not require additional treatment.

"In this talk I will present some recent developments obtained on structure-preserving FEEC schemes. The first one is a general study of commuting de Rham diagrams with broken FEEC spaces on multi-patch domains, allowing for local (single patch) L2 projections and local commuting projections, as well as for domains with non uniform grids. The second development is an extension of this approach which allows to smoothly discretize polar domains with regular tensor-product splines. If time allows, I will finally present a structure-preserving method for solving exterior magnetic problems with constraints of circulation type imposed on strongly divergence-free fields."

This work presents a fundamental advancement in algebraic Variational Multiscale (VMS) methods through the employment of dual basis functions. We demonstrate how dual basis functions enable the computation of projections and the explicit construction of the Fine-Scale Greens' function associated to a projector. We show that the Fine-Scale Greens' functions computed via the dual functions accurately reconstruct the unresolved scales truncated by the projector thereby enabling an algebraic treatment of the unresolved scales. Additionally, we analyse the structure of optimal projectors associated with various partial differential equations, providing further insight into their role in multiscale modelling. Building on this foundation, we propose an algebraic VMS approach that incorporates the Fine-Scale Greens' function, yielding a numerical solution that closely approximates the chosen projection of the continuous solution. We showcase the framework for 1D and 2D advection-diffusion problems as well as the 2D incompressible Navier-Stokes equations, demonstrating its ability to systematically approximate fine-scale effects. Furthermore, we highlight an application of the VMS approach to mesh adaptation (r-adaptation), where computational nodes are dynamically redistributed to capture local solution features based on the computed unresolved scales.

Open discussion on the topic of Structure-Preserving Spatial Discretizations. - Discuss content of presentations - Relevant future developments - Challenges - etc.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The purpose of this talk is twofold. The first is to provide a general introduction to structure-preserving time-integration techniques, from the foundations of the field to some recent research directions. The second is to discuss some of my own contributions over the last few years, focusing particularly on the concept of "functional equivariance."

In this talk, I will present a structure-preserving discretization of one-dimensional thermal-fluid models using the metriplectic 4-bracket formalism. The metriplectic framework extends Hamiltonian mechanics to include dissipation while satisfying the first and second laws of thermodynamics. I will discuss how the 4-bracket can be leveraged to design Galerkin spatial discretizations which automatically retain the symmetries and degeneracies of the 4-bracket. This yields spatial semi-discretizations which are energy-conserving and entropy-producing. Temporal discretization via energy-preserving discrete gradient methods yields a thermodynamically consistent fully discrete scheme. Numerical examples of one-dimensional compressible flow illustrate the approach and confirm the expected behavior. This work suggests a broader applicability of metriplectic 4-brackets in the design of thermodynamically consistent numerical methods.

Variational principles are powerful tools for the modelling and simulation of conservative mechanical and electrical systems. As it is well-known, the fulfilment of a variational principle leads to the Euler-Lagrange equations of motion describing the dynamics of such systems. A discretisation of the variational principle leads to unified numerical schemes called variational integrators with powerful structure-preserving properties such as symplecticity, momentum preservation and excellent long-time behaviour. After a broad introduction to variational integrators we will focus on different recent research aspects. These include a multi rate version for the efficient simulation of dynamics on different time scales as well as an extension towards dissipative systems based on fractional variational principles. The theoretical results will be demonstrated numerically by means of several applications.

In this talk, I will present an overview of our group's recent work at the intersection of continuum mechanics, geometric numerical integration, and optimal control. Our primary focus has been on beams and the challenges of incorporating them into optimal control problems, with particular attention to the geometrically exact beam (GEB). We start by exploring a simplified beam model, applying a modal decomposition approach and multi-rate approximations within the Discrete Mechanics and Optimal Control (DMOC) framework. Next, we address the challenges posed by the numerical integration of GEBs and highlight our contributions in this area. We then extend our discussion to an optimal control problem, using an enhanced version of DMOC that incorporates constraints through discrete augmented cost functions. Finally, I will introduce a new Lagrangian approach to optimal control we have recently developed, which holds promise for future advancements in this field.

Open discussion on the topic of Structure-Preserving Time Integrators. - Discuss content of presentations - Relevant future developments - Challenges - etc.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

What makes good scientific software? Accuracy, stability, and efficiency are obviously necessary attributes, but are they sufficient? How do we design software so that it is understandable and usable by others, able to be ported and optimized for new architectures, able to be extended to new uses and developed by unrelated groups? We will give examples from structure-preserving methods to motivate these design principles.

This talk will introduce a new FEM Julia package: Mantis.jl. This package was designed to discretise the differential geometry formulation of PDEs, focussing on structure preservation. Therefore, Mantis supports differential forms and their operators (e.g. the exterior derivative, Hodge-star, and wedge) and various FEM function spaces. These operators and function spaces are dimension-independent and defined over general manifolds of dimensions 1 to 3. Manifolds can be defined via user-defined expressions or approximated via function spaces. Users can extend Mantis by adding anything from new basis functions to weak formulations to solvers. Some examples will be presented, and how Mantis is used in these cases will be described. These examples will highlight Mantis' versatility by looking at the Hodge-Laplacian, Maxwell-eigenvalue problem, and Stokes' flow, using mimetic spectral element basis functions, single- and multi-patch splines, and local refinement. The future development plans will also be briefly discussed.

Modern GPU-based exascale architectures require rethinking of the numerical algorithms used in large-scale PDE-based applications. These architectures favor algorithms, such as high-order finite elements, that expose fine-grain parallelism and maximize the ratio of floating-point operations to energy intensive data movement. In this talk we present an overview of MFEM (https://mfem.org), a scalable library for high-order finite element discretization of PDEs on general unstructured grids that employs partial assembly and matrix-free algorithms to power a wide variety of HPC applications. Our approach to efficient operator evaluation is based on a decomposed representation of the finite element operator, that factors a bilinear form into a series of sparse and dense components corresponding to the parallelism, mesh topology, basis, geometry, and pointwise physics in the problem. This exposes several layers of parallelism, enables the use of batched dgemss and tensor contractions, and only requires quadrature point values to be assembled for computing the action. The "partial assembly" formulation is a natural fit for modern HPC hardware because it results both in less (nearly optimal) computation and less (optimal) data movement compared to assembling a global sparse matrix, therefore increasing performance and reducing time to solution. In addition to discussing MFEM's capabilities and algorithms, and demonstrate their impact in several large-scale applications from the US Department of Energy.

Struphy (STRUcture-Preserving HYbrid codes) provides easy access to modules for solving PDEs with geometric methods for the purpose of teaching, research and HPC applications. Struphy's Python API allows integration with other codes and provides a convenient interface for learning code objects in small programs or Jupyter notebooks. It follows the paradigm "from proto-typing to production in a single framework". Current numerical methods include FEEC (finite element exterior calculus), PIC (particle-in-cell) and purely Lagrangian methods (such as smoothed-particle hydrodynamics). All Struphy objects can be used in 1d, 2d, or 3d (plus corresponding velocity space for kinetic objects), on arbitrary mapped domains (single patch at the moment) and with MPI parallelization. GPU acceleration of computing kernels is on the way. The aim of the project is to provide researchers with a platform to easily implement and showcase new algorithms in realistic settings.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Intrepid2 is a package within Trilinos that provides a host of tools for finite element and related discretizations. In this talk we focus particularly on the De Rham-conforming basis functions provided by Intrepid2, and template mechanisms for easily extending these to other (possibly exotic) bases by simply providing 1D H(grad) and L^2 basis functions.

Conventional machine learning approaches fail to properly encode structures that provide guarantees for structure preservation that compatible discretizations provide. To remedy this, we introduce a framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. We show a range of results, focusing on building surrogate models for electron transport pathways through a lithium-ion microstructure; for this problem, we reduce the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

My talk comprises two parts. The first problem we consider is as follows. Suppose we observe a system that we know is Lie--Poisson, and we know the bracket exactly. However, we do not know the exact physics behind the problem, so we don't know the Hamiltonian behind the problem. It is shown that the solution to the problem can be found based on observations only. The second problem we discuss is the following. Suppose we observe a set of interacting objects that are performing some motion, and each of these objects evolves on a Lie group. Suppose also that the objects are controlled, and we have some information about the control procedure. An example of such a system is a flock of birds, or, for more industrial applications, a set of interacting drones. As it turns out, the resulting dynamics and control can also be written as Lie--Poisson control system. We prove that we can predict the dynamics of the system based exclusively on the observations.

Hamiltonian systems represent idealized physical models with important properties. Their extensions to metriplectic, entropy generating dynamical systems are equally important, capturing a wide variety of realistic and interesting physics. We discuss recent progress in the structure-preserving model reduction and surrogate modeling of these systems, with emphasis on stability, dynamical discovery, and predictive capability.

In the last decades, different fields emerged to discretize and simulate physical problems in a structure-preserving, efficient and stable manner (DEC, FEEC, FES, DDG). On the other hand, geometric mechanics (GM) became a powerful framework for understanding and designing complex materials (e.g. Cosserat). The advances of both groups rely on the same techniques from modern differential geometry (DG) and algebraic topology (AT), with numerical analysis bridging the gap: simplicial complexes extend discrete topology to continuous fields. Tensor calculus on Riemannian manifolds, with a connection given by covariant differentiation in terms of a metric, has long been a powerful tool in classical mechanics. However, for the description of generalized materials and dislocation theory, where the manifold describing the location of the material is different from the manifold for the additional degrees of freedom, one is left with only tangent frames and differential forms (with values in a vector bundle). Exterior calculus expresses the derivative operators (div, grad, curl) in terms of the more fundamental wedge product, exterior derivative, and Hodge star operators which separate topology from metric information. De Rham cohomology, the link between differential forms and topology, has led to discretizations that preserve the integral theorems of simplicial complexes. From DEC we know that pairing values on cochains with chains is the discrete analogue of integration of a continuous scalar-valued differential form over a domain. FEEC used cohomology and Hodge theory to derive a unified framework and build a discrete de Rham complex that is isomorphic to its continuous version. FES combines FEEC with DDG to construct discrete spaces which form a subcomplex of certain Hilbert complexes, which allows describing e.g. Cosserat elasticity and linearized Riemann–Cartan geometry. However, generalizing the discretization of scalar-valued differential forms on a manifold which is now well-understood, to the discretization of differential forms with values in a vector-bundle with a connection is notoriously difficult and requires advanced concepts from functional analysis, modern differential geometry, algebraic topology, and numerical mathematics. From a maintainer's perspective, supporting both GM and SPD, this talk will illustrate the story above and highlight key elements.

Open discussion on the topic of Software Frameworks and the Role of AI in Scientific Computation. - Discuss content of presentations - Relevant future developments - Challenges - etc.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Open discussion on the Future Steps - Future workshops - Dedicate conference - Joint research proposals

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.