arXiv subjects

Marc C. Steinbach

Publications and source records attributed to Marc C. Steinbach.

17 recordsLinked to original sources

arXiv2026

Intrinsic Structure of Elasticity in Hilbert Space

Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.

math.AP↗

arXiv2026

Numerical Analysis of a Cut Finite Element Approach for Fully Eulerian Fluid-Structure Interaction with Fixed Interface

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.

math.NA↗

arXiv2025

Accurate Thyroid Cancer Classification using a Novel Binary Pattern Driven Local Discrete Cosine Transform Descriptor

In this study, we develop a new CAD system for accurate thyroid cancer classification with emphasis on feature extraction. Prior studies have shown that thyroid texture is important for segregating the thyroid ultrasound images into different classes. Based upon our experience with breast cancer classification, we first conjuncture that the Discrete Cosine Transform (DCT) is the best descriptor for capturing textural features. Thyroid ultrasound images are particularly challenging as the gland is surrounded by multiple complex anatomical structures leading to variations in tissue density. Hence, we second conjuncture the importance of localization and propose that the Local DCT (LDCT) descriptor captures the textural features best in this context. Another disadvantage of complex anatomy around the thyroid gland is scattering of ultrasound waves resulting in noisy and unclear textures. Hence, we third conjuncture that one image descriptor is not enough to fully capture the textural features and propose the integration of another popular texture capturing descriptor (Improved Local Binary Pattern, ILBP) with LDCT. ILBP is known to be noise resilient as well. We term our novel descriptor as Binary Pattern Driven Local Discrete Cosine Transform (BPD-LDCT). Final classification is carried out using a non-linear SVM. The proposed CAD system is evaluated on the only two publicly available thyroid cancer datasets, namely TDID and AUITD. The evaluation is conducted in two stages. In Stage I, thyroid nodules are categorized as benign or malignant. In Stage II, the malignant cases are further sub-classified into TI-RADS (4) and TI-RADS (5). For Stage I classification, our proposed model demonstrates exceptional performance of nearly 100% on TDID and 97% on AUITD. In Stage II classification, the proposed model again attains excellent classification of close to 100% on TDID and 99% on AUITD.

cs.CV↗

arXiv2025

Numerical Simulations of Fully Eulerian Fluid-Structure Contact Interaction using a Ghost-Penalty Cut Finite Element Approach

In this work, we develop a cut-based unfitted finite element formulation for solving nonlinear, nonstationary fluid-structure interaction with contact in Eulerian coordinates. In the Eulerian description fluid flow modeled by the incompressible Navier-Stokes equations remains in Eulerian coordinates, while elastic solids are transformed from Lagrangian coordinates into the Eulerian system. A monolithic description is adopted. For the spatial discretization, we employ an unfitted finite element method with ghost penalties based on inf-sup stable finite elements. To handle contact, we use a relaxation of the contact condition in combination with a unified Nitsche approach that takes care implicitly of the switch between fluid-structure interaction and contact conditions. The temporal discretization is based on a backward Euler scheme with implicit extensions of solutions at the previous time step. The nonlinear system is solved with a semi-smooth Newton's method with line search. Our formulation, discretization and implementation are substantiated with an elastic falling ball that comes into contact with the bottom boundary, constituting a challenging state-of-the-art benchmark.

math.NA↗

arXiv2024

Optimizing Non-Intersecting Synthetic Vascular Trees in Nonconvex Organs

The understanding of the mechanisms driving vascular development is still limited. Techniques to generate vascular trees synthetically have been developed to tackle this problem. However, most algorithms are limited to single trees inside convex perfusion volumes. We introduce a new framework for generating multiple trees inside general nonconvex perfusion volumes. Our framework combines topology optimization and global geometry optimization into a single algorithmic approach. Our first contribution is defining a baseline problem based on Murray's original formulation, which accommodates efficient solution algorithms. The problem of finding the global minimum is cast into a nonlinear optimization problem (NLP) with merely super-linear solution effort. Our second contribution extends the NLP to constrain multiple vascular trees inside any nonconvex boundary while avoiding intersections. We test our framework against a benchmark of an anatomic region of brain tissue and a vasculature of the human liver. In all cases, the total tree energy is improved significantly compared to local approaches. By avoiding intersections globally, we can reproduce key physiological features such as parallel running inflow vessels and tortuous vessels. The ability to generate non-intersecting vascular trees inside nonconvex organs can improve the functional assessment of organs.

physics.bio-ph↗

arXiv2024

Damage Location in Mechanical Structures by Multi-Objective Pattern Search

We propose a multi-objective global pattern search algorithm for the task of locating and quantifying damage in flexible mechanical structures. This is achieved by identifying eigenfrequencies and eigenmodes from measurements and matching them against the results of a finite element simulation model, which leads to a nonsmooth nonlinear bi-objective parameter estimation problem. A derivative-free optimization algorithm is required since the problem is nonsmooth and also because complex mechanical simulation models are often solved using commercial black-box software. Moreover, the entire set of non-dominated solutions is of interest to practitioners. Most solution approaches published to date are based on meta-heuristics such as genetic algorithms. The proposed multi-objective pattern-search algorithm provides a mathematically well-founded alternative. It features a novel sorting procedure that reduces the complexity in our context. Test runs on two experimental structures with multiple damage scenarios are used to validate the approach. The results demonstrate that the proposed algorithm yields accurate damage locations and requires moderate computational resources. From the engineer's perspective it represents a promising tool for structural health monitoring.

math.OC↗

arXiv2024

Accelerating Aeroelastic UVLM Simulations by Inexact Newton Algorithms

We consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based in this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations.

math.NA↗

arXiv2024

Modeling and numerical simulation of fully Eulerian fluid-structure interaction using cut finite elements

We present a monolithic finite element formulation for (nonlinear) fluid-structure interaction in Eulerian coordinates. For the discretization we employ an unfitted finite element method based on inf-sup stable finite elements. So-called ghost penalty terms are used to guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The resulting system is solved in a monolithic fashion using Newton's method. Our developments are tested on a numerical example with fixed interface.

math.NA↗

arXiv2023

Analysis of a space-time phase-field fracture complementarity model and its optimal control formulation

The purpose of this work is the formulation of optimality conditions for phase-field optimal control problems. The forward problem is first stated as an abstract nonlinear optimization problem, and then the necessary optimality conditions are derived. The sufficient optimality conditions are also examined. The choice of suitable function spaces to ensure the regularity of the nonlinear optimization problem is a true challenge here. Afterwards the optimal control problem with a tracking type cost functional is formulated. The constraints are given by the previously derived first order optimality conditions of the forward problem. Herein regularity is proven under certain conditions and first order optimality conditions are formulated.

math.OC↗

arXiv2023

Branching exponents of synthetic vascular trees under different optimality principles

The branching behavior of vascular trees is often characterized using Murray's law. We investigate its validity using synthetic vascular trees generated under global optimization criteria. Our synthetic tree model does not incorporate Murray's law explicitly. Instead, we assume it holds implicitly and investigate the effects of different physical constraints and optimization goals on the branching exponent that is now allowed to vary locally. In particular, we include variable blood viscosity due to the Fåhræus--Lindqvist effect and enforce an equal pressure drop between inflow and the micro-circulation. Using our global optimization framework, we generate vascular trees with over one million terminal vessels and compare them against a detailed corrosion cast of the portal venous tree of a human liver. Murray's law is implicitly fulfilled when no additional constraints are enforced, indicating its validity in this setting. Variable blood viscosity or equal pressure drop leads to deviations from this optimum, but with the branching exponent inside the experimentally predicted range between 2.0 and 3.0. The validation against the corrosion cast shows good agreement from the portal vein down to the venules. Not enforcing Murray's law explicitly reduces the computational cost and increases the predictive capabilities of synthetic vascular trees. The ability to study optimal branching exponents across different scales can improve the functional assessment of organs.

physics.bio-ph↗

arXiv2022

Principal Geodesic Analysis in Director-Based Dynamics of Hybrid Mechanical Systems

In this article, we present new computational realizations of principal geodesic analysis for the unit sphere $S^2$ and the special orthogonal group $SO(3)$. In particular, we address the construction of long-time smooth lifts across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical precision. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algortihms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical beahvior.

math.DS↗

arXiv2022

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

math.OC↗

arXiv2022

SABMIS: Sparse approximation based blind multi-image steganography scheme

We hide grayscale secret images into a grayscale cover image, which is considered to be a challenging steganography problem. Our goal is to develop a steganography scheme with enhanced embedding capacity while preserving the visual quality of the stego-image as well as the extracted secret image, and ensuring that the stego-image is resistant to steganographic attacks. The novel embedding rule of our scheme helps to hide secret image sparse coefficients into the oversampled cover image sparse coefficients in a staggered manner. The stego-image is constructed by using ADMM to solve the LASSO formulation of the underlying minimization problem. Finally, the secret images are extracted from the constructed stego-image using the reverse of our embedding rule. Using these components together, to achieve the above mentioned competing goals, forms our most novel contribution. We term our scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography). We perform extensive experiments on several standard images. By choosing the size of the secret images to be half of the of cover image, we obtain embedding capacities of 2 bpp (bits per pixel), 4 bpp, 6 bpp, and 8 bpp while embedding one, two, three, and four secret images, respectively. Our focus is on hiding multiple secret images. For the case of hiding two and three secret images, our embedding capacities are higher than all the embedding capacities obtained in the literature until now. For the case of hiding four secret images, although our capacity is slightly lower than one work, we do better on the other two goals; a) very little deterioration in the quality of the stego-images and extracted secret images, and b) inherently and designed-to-be resistant to steganographic attacks. Additionally, we demonstrate that SABMIS executes in few minutes, and show its application on two real-life problems.

cs.CR↗

arXiv2022

Space-time formulation, discretization, and computations for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization termsand the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Our algorithmic developments are substantiated and illustrated with some numerical experiments.

math.OC↗

arXiv2022

Rigorous mathematical optimization of synthetic hepatic vascular trees

In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure to the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.

physics.bio-ph↗

arXiv2021

Relations between Abs-Normal NLPs and MPCCs. Part 1: Strong Constraint Qualifications

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then we consider strong stationarity concepts with first and second order optimality conditions, which again turn out to be equivalent for the two problem classes. Throughout we also consider specific slack reformulations suggested in [9], which preserve constraint qualifications of linear independence type but not of Mangasarian-Fromovitz type.

math.OC↗

arXiv2021

Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.

math.OC↗