Adrián J. Lew

Maurizio M. Chiaramonte, Yongxing Shen, Leon M. Keer et al.

The use of the interaction integral to compute stress intensity factors around a crack tip requires selecting an auxiliary field and a material variation field. We formulate a family of these fields accounting for the curvilinear nature of cracks that, in conjunction with a discrete formulation of the interaction integral, yield optimally convergent stress intensity factors. We formulate three pairs of auxiliary and material variation fields chosen to yield a simple expression of the interaction integral for different classes of problems. The formulation accounts for crack face tractions and body forces. Distinct features of the fields are their ease of construction and implementation. The resulting stress intensity factors are observed converging at a rate that doubles the one of the stress field. We provide a sketch of the theoretical justification for the observed convergence rates, and discuss issues such as quadratures and domain approximations needed to attain such convergent behavior. Through two representative examples, a circular arc crack and a loaded power function crack, we illustrate the convergence rates of the computed stress intensity factors. The numerical results also show the independence of the method on the size of the domain of integration.

Ramsharan Rangarajan, Adrian J. Lew

We prove that a planar $C^2$-regular boundary $Γ$ can always be parameterized with its closest point projection $π$ over a certain collection of edges $Γ_h$ in an ambient triangulation, by making simple assumptions on the background mesh. For $Γ_h$, we select the edges that have both vertices on one side of $Γ$ and belong to a triangle that has a vertex on the other side. By imposing restrictions on the size of triangles near the curve and by requesting that certain angles in the mesh be strictly acute, we prove that $π:Γ_h\rightarrowΓ$ is a homeomorphism, that it is $C^1$ on each edge in $Γ_h$ and provide bounds for the Jacobian of the parameterization. The assumptions on the background mesh are both easy to satisfy in practice and conveniently verified in computer implementations. The parameterization analyzed here was previously proposed by the authors and applied to the construction of high-order curved finite elements on a class of planar piecewise $C^2$-curves.

Ramsharan Rangarajan, Adrian J. Lew

We describe a method for discretizing planar C2-regular domains immersed in non-conforming triangulations. The method consists in constructing mappings from triangles in a background mesh to curvilinear ones that conform exactly to the immersed domain. Constructing such a map relies on a novel way of parameterizing the immersed boundary over a collection of nearby edges with its closest point projection. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the background mesh. Indeed, interpolating the constructed mappings just at the vertices of the background mesh yields a fast meshing algorithm that involves only perturbing a few vertices near the boundary. For the discretization of a curved domain to be robust, we have to impose restrictions on the background mesh. Conversely, these restrictions define a family of domains that can be discretized with a given background mesh. We then say that the background mesh is a universal mesh for such a family of domains. The notion of universal meshes is particularly useful in free/moving boundary problems because the same background mesh can serve as the universal mesh for the evolving domain for time intervals that are independent of the time step. Hence it facilitates a framework for finite element calculations over evolving domains while using a fixed background mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. We demonstrate these ideas with various numerical examples.

Evan S. Gawlik, Adrian J. Lew

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in particular, to the setting in which the two finite element spaces consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure $O(h^γ)$, where $γ$ is a nonnegative scalar and $h$ is the mesh spacing. The projector may be, for example, the orthogonal projector with respect to the $L^2$- or $H^1$-inner product. In these and other circumstances, the bounds are superconvergent under a few mild regularity assumptions. That is, under mesh refinement, the two projections differ in norm by an amount that decays to zero at a faster rate than the amounts by which each projection differs from $u$. We present numerical examples to illustrate these superconvergent estimates and verify the necessity of the regularity assumptions on $u$.

Mustafa Hajij, Lennart Bastian, Sarah Osentoski et al.

We introduce copresheaf topological neural networks (CTNNs), a powerful unifying framework that encapsulates a wide spectrum of deep learning architectures, designed to operate on structured data, including images, point clouds, graphs, meshes, and topological manifolds. While deep learning has profoundly impacted domains ranging from digital assistants to autonomous systems, the principled design of neural architectures tailored to specific tasks and data types remains one of the field's most persistent open challenges. CTNNs address this gap by formulating model design in the language of copresheaves, a concept from algebraic topology that generalizes most practical deep learning models in use today. This abstract yet constructive formulation yields a rich design space from which theoretically sound and practically effective solutions can be derived to tackle core challenges in representation learning, such as long-range dependencies, oversmoothing, heterophily, and non-Euclidean domains. Our empirical results on structured data benchmarks demonstrate that CTNNs consistently outperform conventional baselines, particularly in tasks requiring hierarchical or localized sensitivity. These results establish CTNNs as a principled multi-scale foundation for the next generation of deep learning architectures.