Speakers

Computational Medicine is a rapidly growing field. I would divide the history of Computational Medicine into two eras, one prior to the widespread use of medical imaging modalities and one since. In the former era, Computational Medicine was in its infancy and the relevance of results, often computed on very simplified geometric configuration, was limited and had very little impact on clinical practice. Since the clinical installation of imaging technology, the situation has changed significantly and continues to progress. The early days of Computational Mechanics saw a similar evolution, but as computers became more powerful and ubiquitous in engineering

design offices, the footprint of Computational Mechanics became pervasive. Problems that could not be solver early on, such as full vehicle crash analysis in the 1980s, became routine by the late 1990s, and now we can say that cars are fully designed on computers.

The fidelity of imaging modalities continues to advance and seems to behave similarly to the way Moore’s Law did for the number of transistors on a microchip. It has been said that now Moore’s Law is now at work in medical imaging, and it is clearly a driver for the development and application of Computational Medicine. The future of Computational Medicine may look like the present of Computational Mechanics, which also continues to move forward.

The problems facing Computational Medicine are the complexity of organs and physiological processes, the current resolution of imaging modalities with reference to particular diseases, the relevant time scales to affect patient specific clinical decision making, and how to deliver the results of Computational Medicine analyses to the clinician. The major technical pacing item is the accurate and efficient creation of computational models, which is also still the major bottleneck in the engineering design-through-analysis process. Clearly, Isogeometric Analysis has a role to play in addressing the image-based modeling problem of Computational Medicine. In engineering, the predominant modeling problem is building models from Computer Aided Design (CAD) files. A question that is at present unanswered is should a CAD file first be created of the organs under consideration, or should the computational model be directly built from the segmentation of the organs, e.g., the coronary arteries, cancerous tumors, etc.? Another question is where does AI and Machine Learning fit into the process of developing computational models from imaging data? These technologies are already of use in some clinically available technologies, e.g., HeartFlow, Inc.

I do not have answers to all the questions, but I do have examples of how Isogeometric Analysis is being used in Computational Medicine, both from the model building and equation solving perspectives. In my presentation, I will focus on recent Isogeometric Analysis work on the detection risk-assessment of vulnerable plaques in the walls of coronary arteries, but I will also briefly draw attention to the active surveillance and management of prostate tumors, and initiatory efforts to model glymphatic transport and amyloid plaque deposition. The latter topic is the subject of a presentation by Shaolie Hossain during this conference. I will also present examples of boundary-fitted and immersed model building technologies, and a way to model

essential sub-voxel (i.e., “invisible”) features. The immersed approach seems to have significant advantages in many Computational Medicine scenarios.

In my opinion, research opportunities going forward abound for Isogeometric Analysis and Computational Medicine.

The mathematical foundations of adaptive methods for the numerical solution of partial differential equations have been widely studied for several decades. Starting from a given initial mesh, the aim is to increase the accuracy of a discrete solution by iterating the building blocks of the so-called adaptive loop. At each refinement step, the discrete solution on the current mesh is derived, local contributions of some a posteriori error estimator are computed, a set of mesh elements are marked for refinement/coarsening, and the new mesh for the next iterative step is generated by refining/coarsening (at least) all marked elements. It should be noted, however, that in spite of such a long history of adaptivity theory, the application of adaptive methods in 3D is often very complex and expensive. In particular, it naturally poses several challenges, like, e.g., the development of a numerical scheme able to effectively harmonize adaptivity in complex physical models, the implementation of robust discretizations guaranteeing a suitable trade-off between computational accuracy and number of degrees of freedom, and the treatment of complex geometries. This talk will address these issues by presenting an overview of recent advances in the design and analysis of adaptive isogeometric methods with special focus on standard and novel hierarchical spline constructions, their applications in engineering problems of relevant interest, as well as related extensions in different directions.

Convolution Hierarchical Deep-learning Neural Network (C-HiDeNN) and C-HiDeNN – Tensor Decomposition (C-HiDeNN-TD) is introduced for modeling and controlling of Laser Powder Bed Fusion (L-PBF) process in metal additive manufacturing (AM) [1]. Instead of using machine learning models as black boxes, the C-HiDeNN [2] that leverages the concept of interpolation and tensor decomposition (TD) is developed. C-HiDeNN and C-HiDeNN-TD [3, 4] unifies training, solving, and calibration, and achieves orders of magnitude less trainable parameters (or degrees of freedom), faster training/solving, smaller memory footprint, and higher model accuracy compared to feed-forward neural networks and physics-informed neural networks. C-HiDeNN-TD facilitates extremely challenging AM process simulations (over Ze[a-scale, ) and online control. To analyze additively manufactured parts predefined with complex computer-aided design (CAD) geometry, we propose immersed methods combined with C-HiDeNN-TD that enables a mesh up to nodes. This work is motivated by multi-patch IsoGeometric (IGA) theories [5-8]. We anticipate the proposed C-HiDeNN and C-HiDeNN-TD will assist both manufacturers of additive manufacturing equipment and end users in optimizing the processing parameters and manufactured components and systems. The performance of C-HiDeNN-TD can further be enhanced by the groundbreaking work variational multiscale method (VMS) developed by TJR Hughes in 1998.

Trimmed or cut isogeometric analysis involves embedding a computational domain into a background mesh that does not conform to the domain's boundary, resulting in trimmed elements at the boundary. One approach adds stabilization terms to the weak formulation to achieve stability near the boundary, controlling the variation of discrete functions. This method allows us to prove stability, estimate condition numbers, and obtain optimal-order a priori error estimates. Alternatively, we can use a discrete extension operator to solve the problem within a subspace of the finite element space, eliminating unstable degrees of freedom while retaining optimal approximation bounds.

Although these two approaches—the weak addition of stabilization terms and the strong enforcement via extension operators—seem different, they share the common goal of stabilizing the method. We demonstrate that the definition of stabilization terms added to the weak statement can be generalized in two significant ways:

1. Stabilized Quantities: The quantity being stabilized can be any functional of the discrete function, such as finite element degrees of freedom. This allows for more precise stabilization of unstable modes than standard, element-based methods.

2. Choice of Connected Elements: Instead of using face neighbors or connected patches, we can stabilize by connecting elements that intersect the boundary to elements within a distance proportional to the mesh parameter.

We show that this generalized stabilization framework fits within standard abstract requirements, leading to stable and optimally convergent methods for second-order elliptic problems. Additionally, we demonstrate that for a robust design of the ghost penalty, the stabilization parameter can tend to infinity without causing locking. In this limit, the stabilization corresponds to the strong enforcement of certain algebraic constraints, identical to those implemented in specific extension operator frameworks. This observation reveals a close connection between stabilization and extension approaches.

Several application examples are presented to illustrate these concepts.

This presentation explores the application of Isogeometric Analysis (IGA) in conjunction with the Variational Multiscale (VMS) formulation for modeling environmental flows. We address the Navier-Stokes equations for incompressible stratified flows with the Boussinesq approximation, incorporating an additional advection-diffusion equation for the temperature field. The methodology employs NURBS-based discretization, allowing for superior geometric flexibility and higher-order continuity. The framework is further enhanced with weak imposition of Dirichlet boundary conditions and wall modeling to capture the effects of surface roughness.

The proposed approach is validated through multiple benchmark cases, including stratified channel flow and flow over 3D hills with complex topography. Comparisons with field data, laboratory experiments, and high-fidelity simulations like DNS and LES demonstrate the framework's ability to accurately capture phenomena such as intermittent turbulence, low-level jets, and flow separation. The results underscore the potential of this IGA-based VMS framework for a wide range of environmental flow applications, including atmospheric boundary layer modeling and renewable energy applications like wind farm simulations.

Since the initial development of isogeometric analysis (IGA) and its evolution over the last two decades, research on IGA methods has advanced significantly, leading to a well-established and continuously expanding field. Alongside the growth of IGA, the computational engineering field has also seen significant improvements in computer-aided design (CAD) and computer-aided engineering (CAE) technologies and methods. Despite these advances and the movement toward digital thread environments in many areas of industry that benefit significantly from comprehensively integrated design, manufacturing, and operating procedures, standard design practices often remain the primary strategy for many engineering applications. The presented developments offer novel approaches to reimagining the engineering design process through interconnected computational design and analysis infrastructures that leverage parametric and algorithmic modeling. These methods are augmented by the fundamental premise of IGA, which enables direct integration of design and analysis. Several industrial applications are presented to demonstrate the proposed parametric and algorithmic modeling methods, including biomedical and aerospace applications. These strategies enable the development of novel design families or configurations that can be flexibly modified and adjusted to accommodate varying constraints or operating conditions.