arXiv subjects

Daniel Balzani

Publications and source records attributed to Daniel Balzani.

15 recordsLinked to original sources

arXiv2025

Homeostatic Kinematic Growth Model for Arteries -- Residual Stresses and Active Response

A simple kinematic growth model for muscular arteries is presented which allows the incorporation of residual stresses such that a homeostatic in-vivo stress state under physiological loading is obtained. To this end, new evolution equations for growth are proposed, which avoid issues with instability of final growth states known from other kinematric growth models. These evolution equations are connected to a new set of growth driving forces. By introducing a formulation using the principle Cauchy stresses, reasonable in vivo stress distributions can be obtained while ensuring realistic geometries of arteries after growth. To incorporate realistic Cauchy stresses for muscular arteries under varying loading conditions, which appear in vivo, e.g., due to physical activity, the growth model is combined with a newly proposed stretch-dependent model for smooth muscle activation. To account for the changes of geometry and mechanical behavior during the growth process, an optimization procedure is described which leads to a more accurate representation of an arterial ring and its mechanical behavior after the growth-related homogenization of the stresses is reached. Based thereon, parameters are adjusted to experimental data of a healthy middle cerebral artery of a rat showing that the proposed model accurately describes real material behavior. The successful combination of growth and active response indicates that the new growth model can be used without problems for modeling active tissues under various conditions.

q-bio.QM↗

arXiv2025

A Simple Voltage-Modulated Markov Chain Model for the Piezo1 Ion Channel to Investigate Electromechanical Pacing

Piezo1 ion channels are voltage-modulated, stretch-activated ion channels involved in a variety of important physiological and pathophysiological processes, as for example cardiovascular development and homeostasis. Since its discovery, it has been known that this type of ion channel desensitizes when exposed to stretch. However, recent experiments on Piezo1 ion channels have uncovered that their stretch response is qualitatively different when exposed to positive electrochemical driving forces, where the desensitization is reset. In this work, we propose a novel voltage-modulated mathematical model of Piezo1 based on a continuous-time Markov chain. We show that our Piezo1 model is able to quantitatively reproduce a wide range of experimental observations. Furthermore, we integrate our new ion channel model into the Mahajan-Shiferaw ventricular cardiomyocyte model to study the effect of electromechanical pacing at the cellular scale. This integrated cell model is able to qualitatively reproduce some aspects of the experimental observations regarding the rate-dependence of electromechanical pacing protocols. Our studies suggest that the Piezo1 ion channel is an important component that significantly contributes to the electromechanical coupled response of cardiomyocytes.

q-bio.QM↗

arXiv2025

Quasiconvex relaxation of planar Biot-type energies and the role of determinant constraints

We derive the quasiconvex relaxation of the Biot-type energy density $\lVert\sqrt{\operatorname{D}\varphi^T \operatorname{D}\varphi}-I_2\rVert^2$ for planar mappings $\varphi\colon\mathbb{R}^2\to \mathbb{R}^2$ in two different scenarios. First, we consider the case $\operatorname{D}\varphi\in\textrm{GL}^+(2)$, in which the energy can be expressed as the squared Euclidean distance $\operatorname{dist}^2(\operatorname{D}\varphi,\textrm{SO}(2))$ to the special orthogonal group $\textrm{SO}(2)$. We then allow for planar mappings with arbitrary $\operatorname{D}\varphi\in\mathbb{R}^{2\times 2}$; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self-interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank-one sequential lamination algorithm, trust-region FEM calculations of representative microstructures and physics-informed neural networks.

math.AP↗

arXiv2024

Reliability-Based Design Optimization Incorporating Extended Optimal Uncertainty Quantification

Reliability-based design optimization (RBDO) approaches aim to identify the best design of an engineering problem, whilst the probability of failure (PoF) remains below an acceptable value. Thus, the incorporation of the sharpest bounds on the PoF under given constraints on the uncertain input quantities strongly strenghtens the significance of RBDO results, since unjustified assumptions on the input quantities are avoided. In this contribution, the extended Optimal Uncertainty Quantification framework is embedded within an RBDO context in terms of a double loop approach. By that, the mathematically sharpest bounds on the PoF as well as on the cost function can be computed for all design candidates and compared with acceptable values. The extended OUQ allows the incorporation of aleatory as well as epistemic uncertainties, where the definition of probability density functions is not necessarily required and just given data on the input can be included. Specifically, not only bounds on the values themselves, but also bounds on moment constraints can be taken into account. Thus, inadmissible assumptions on the data can be avoided, while the optimal design of a problem can be identified. The capability of the resulting framework is firstly shown by means of a benchmark problem under the influence of polymorphic uncertainties. Afterwards, a realistic engineering problem is analyzed, where the positioning of laser-hardened lines within a steel sheet for a car crash structure are optimized.

cs.CE↗

arXiv2024

Hierarchical Rank-One Sequence Convexification for the Relaxation of Variational Problems with Microstructures

This paper presents an efficient algorithm for the approximation of the rank-one convex hull in the context of nonlinear solid mechanics. It is based on hierarchical rank-one sequences and simultaneously provides first and second derivative information essential for the calculation of mechanical stresses and the computational minimization of discretized energies. For materials, whose microstructure can be well approximated in terms of laminates and where each laminate stage achieves energetic optimality with respect to the current stage, the approximate envelope coincides with the rank-one convex envelope. Although the proposed method provides only an upper bound for the rank-one convex hull, a careful examination of the resulting constraints shows a decent applicability in mechanical problems. Various aspects of the algorithm are discussed, including the restoration of rotational invariance, microstructure reconstruction, comparisons with other semi-convexification algorithms, and mesh independency. Overall, this paper demonstrates the efficiency of the algorithm for both, well-established mathematical benchmark problems as well as nonconvex isotropic finite-strain continuum damage models in two and three dimensions. Thereby, for the first time, a feasible concurrent numerical relaxation is established for an incremental, dissipative large-strain model with relevant applications in engineering problems.

cs.CE↗

arXiv2023

An Explicit Local Space-Time Adaptive Framework for Monodomain Models

We present a new explicit local space-time adaptive framework to decrease the time required for monodomain simulations for cardiac electrophysiology. Based on the localized structure of the steep activation wavefront in solutions to monodomain problems, the proposed framework adopts small time steps and a tree-based adaptive mesh refinement scheme only in the regions necessary to resolve these localized structures. The time step and mesh adaptation selection process is fully controlled by a combination of local error indicators. The main contributions of this work consist in the introduction of a primal symmetric interior penalty formulation of the monodomain model and an efficient algorithmic strategy to manage local time stepping for its temporal discretization. In a first serial implementation of this framework, we report decreases in wall-clock time between 2 and 20 times with respect to an optimized implementation of a commonly used numerical scheme, showing that this framework is a promising candidate to accelerate monodomain simulations of cardiac electrophysiology.

math.NA↗

arXiv2023

A computational framework for pharmaco-mechanical interactions in arterial walls using parallel monolithic domain decomposition methods

A computational framework is presented to numerically simulate the effects of antihypertensive drugs, in particular calcium channel blockers, on the mechanical response of arterial walls. A stretch-dependent smooth muscle model by Uhlmann and Balzani is modified to describe the interaction of pharmacological drugs and the inhibition of smooth muscle activation. The coupled deformation-diffusion problem is then solved using the finite element software FEDDLib and overlapping Schwarz preconditioners from the Trilinos package FROSch. These preconditioners include highly scalable parallel GDSW (generalized Dryja-Smith-Widlund) and RDSW (reduced GDSW) preconditioners. Simulation results show the expected increase in the lumen diameter of an idealized artery due to the drug-induced reduction of smooth muscle contraction, as well as a decrease in the rate of arterial contraction in the presence of calcium channel blockers. Strong and weak parallel scalability of the resulting computational implementation are also analyzed.

math.NA↗

arXiv2023

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation by rank-one convexification prevents non-existence of minimizers and mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

cs.CE↗

arXiv2022

A Simple and Efficient Lagrange Multiplier Based Mixed Finite Element for Gradient Damage

A novel finite element formulation for gradient-regularized damage models is presented which allows for the robust, efficient, and mesh-independent simulation of damage phenomena in engineering and biological materials. The paper presents a Lagrange multiplier based mixed finite element formulation for finite strains. Thereby, no numerical stabilization or penalty parameters are required. On the other hand, no additional degrees of freedom appear for the Lagrange multiplier which is achieved through a suitable FE-interpolation scheme allowing for static condensation. In contrast to competitive approaches from the literature with similar efficiency, the proposed formulation does not require cross-element information and thus, a straightforward implementation using standard element routine interfaces is enabled. Numerical tests show mesh-independent solutions, robustness of the solution procedure for states of severe damage and under cyclic loading conditions. It is demonstrated that the computing time of the gradient damage calculations exceeds the one of purely elastic computations only by an insignificant amount. Furthermore, an improved convergence behavior compared to alternative approaches is shown.

cs.CE↗

arXiv2022

Evolving Microstructures in Relaxed Continuum Damage Mechanics for Strain Softening

A new relaxation approach is proposed which allows for the description of stress- and strain-softening at finite strains. The model is based on the construction of a convex hull replacing the originally non-convex incremental stress potential which in turn represents damage in terms of the classical $(1-D)$ approach. This convex hull is given as the linear convex combination of weakly and strongly damaged phases and thus, it represents the homogenization of a microstructure bifurcated in the two phases. As a result thereof, damage evolves in the convexified regime mainly by an increasing volume fraction of the strongly damaged phase. In contrast to previous relaxed incremental formulations in Gürses and Miehe [16] and Balzani and Ortiz [2], where the convex hull has been kept fixated after construction, here, the strongly damaged phase is allowed to elastically unload upon further loading. At the same time, its volume fraction increases nonlinearly within the convexified regime. Thus, strain-softening in the sense of a decreasing stress with increasing strain can be modeled. The major advantage of the proposed approach is that it ensures mesh-independent structural simulations without the requirement of additional length-scale related parameters or nonlocal quantities, which simplifies an implementation using classical material subroutine interfaces. In this paper, focus is on the relaxation of one-dimensional models for fiber damage which are combined with a microsphere approach to allow for the description of three-dimensional fiber dispersions appearing in fibrous materials such as soft biological tissues. Several numerical examples are analyzed to show the overall response of the model and the mesh-independence of resulting structural calculations.

cs.CE↗

arXiv2022

Simulation of crack propagation through voxel-based, heterogeneous structures based on eigenerosion and finite cells

This paper presents an algorithm for the efficient simulation of ductile crack propagation through heterogeneous structures, as e.g. metallic microstructures, which are given as voxel data. These kinds of simulations are required for e.g., the numerical investigation of wear mechanisms at small length scales, which is still a challenging task in engineering. The basic idea of the proposed algorithm is to combine the advantages of the Finite Cell Method allowing for a convenient integration of heterogeneous finite element problems with the eigenerosion approach to still enable the mesh-independent simulation of crack propagation. The major component is to switch from finite subcells to finite elements wherever the crack progresses, thereby automatically adaptively refining at the crack tip by managing the newly appearing nodes as hanging nodes. Technically relevant problems of crack propagation at the microscale are mostly linked with sub-critical crack growth where the crack moves fast and stepwise with subsequent load cycles. Therefore, inertia may become important which is why dynamics are taken into account by spreading the mass of the eroded elements to the nodes to avoid a loss in mass resulting from the erosion procedure. Furthermore, a certain treatment for the finite cell decomposition is considered in order to ensure efficiency and accuracy. The numerical framework as well as the voxel decomposition techniques are analyzed in detail in different three-dimensional numerical examples to show the performance of the proposed approach.

math.NA↗

arXiv2021

A new Method for the in vivo identification of material properties of the human eye. Feasibility analysis based on synthetic data

This paper proposes a new method for in vivo and almost real-time identification of biomechanical properties of the human cornea based on non-contact tonometer data. Further goal is to demonstrate the method's functionality based on synthetic data serving as reference. For this purpose, a finite element model of the human eye is constructed to synthetically generate displacement full-fields from different datasets with keratoconus-like degradations. Then, a new approach based on the equilibrium gap method (EGM) combined with a mechanical morphing approach is proposed and used to identify the material parameters from virtual test data sets. In a further step, random absolute noise is added to the virtual test data to investigate the sensitivity of the new approach to noise. As a result, the proposed method shows a relevant accuracy in identifying material parameters based on displacement full fields. At the same time, the method turns out to work almost in real-time (order of a few minutes on a regular work station) and is thus much faster than inverse problems solved by typical forward approaches. On the other hand, the method shows a noticeable sensitivity to rather small noise amplitudes. However, analysis show that the accuracy is sufficient for the identification of diseased tissue properties.

cs.CE↗

arXiv2021

An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton's principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton's principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton's principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e. displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as principle of virtual work and Onsager's principle, are given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled rate-dependent, rate-independent, and gradient-enhanced materials.

cond-mat.mtrl-sci↗

arXiv2021

Efficient and robust numerical treatment of a gradient-enhanced damage model at large deformations

The modeling of damage processes in materials constitutes an ill-posed mathematical problem which manifests in mesh-dependent finite element results. The loss of ellipticity of the discrete system of equations is counteracted by regularization schemes of which the gradient enhancement of the strain energy density is often used. In this contribution, we present an extension of the efficient numerical treatment, which has been proposed in [1], to materials that are subjected to large deformations. Along with the model derivation, we present a technique for element erosion in the case of severely damaged materials. Efficiency and robustness of our approach is demonstrated by two numerical examples.

cs.CE↗

arXiv2020

A General, Implicit, Large-Strain FE$^2$ Framework for the Simulation of Dynamic Problems on Two Scales

In this paper we present a fully-coupled, two-scale homogenization method for dynamic loading in the spirit of FE$^2$ methods. The framework considers the balance of linear momentum including inertia at the microscale to capture possible dynamic effects arising from micro heterogeneities. A finite-strain formulation is adapted to account for geometrical nonlinearities enabling the study of e.g. plasticity or fiber pullout, which may be associated with large deformations. A consistent kinematic scale link is established as displacement constraint on the whole representative volume element. The consistent macroscopic material tangent moduli are derived including micro inertia in closed form. These can easily be calculated with a loop over all microscopic finite elements, only applying existing assembly and solving procedures. Thus, making it suitable for standard finite element program architectures. Numerical examples of a layered periodic material are presented and compared to direct numerical simulations to demonstrate the capability of the proposed framework.

cs.CE↗