Statistical finite elements for misspecified models.

We present a statistical finite element method for nonlinear, time-dependent phenomena, illustrated in the context of nonlinear internal waves (solitons). We take a Bayesian approach and leverage the finite element method to cast the statistical problem as a nonlinear Gaussian state-space model, updating the solution, in receipt of data, in a filtering framework. The method is applicable to problems across science and engineering for which finite element methods are appropriate. The Korteweg-de Vries equation for solitons is presented because it reflects the necessary complexity while being suitably familiar and succinct for pedagogical purposes. We present two algorithms to implement this method, based on the extended and ensemble Kalman filters, and demonstrate effectiveness with a simulation study and a case study with experimental data. The generality of our approach is demonstrated in SI Appendix, where we present examples from additional nonlinear, time-dependent partial differential equations (Burgers equation, Kuramoto-Sivashinsky equation).

lifex-fiber: an open tool for myofibers generation in cardiac computational models.

BACKGROUND: Modeling the whole cardiac function involves the solution of several complex multi-physics and multi-scale models that are highly computationally demanding, which call for simpler yet accurate, high-performance computational tools. Despite the efforts made by several research groups, no software for whole-heart fully-coupled cardiac simulations in the scientific community has reached full maturity yet. RESULTS: In this work we present [Formula: see text]-fiber, an innovative tool for the generation of myocardial fibers based on Laplace-Dirichlet Rule-Based Methods, which are the essential building blocks for modeling the electrophysiological, mechanical and electromechanical cardiac function, from single-chamber to whole-heart simulations. [Formula: see text]-fiber is the first publicly released module for cardiac simulations based on [Formula: see text], an open-source, high-performance Finite Element solver for multi-physics, multi-scale and multi-domain problems developed in the framework of the iHEART project, which aims at making in silico experiments easily reproducible and accessible to a wide community of users, including those with a background in medicine or bio-engineering. CONCLUSIONS: The tool presented in this document is intended to provide the scientific community with a computational tool that incorporates general state of the art models and solvers for simulating the cardiac function within a high-performance framework that exposes a user- and developer-friendly interface. This report comes with an extensive technical and mathematical documentation to welcome new users to the core structure of [Formula: see text]-fiber and to provide them with a possible approach to include the generated cardiac fibers into more sophisticated computational pipelines. In the near future, more modules will be successively published either as pre-compiled binaries for x86-64 Linux systems or as open source software.

Stability of different fixation methods after reduction malarplasty under average and maximum masticatory forces: a finite element analysis.

BACKGROUND: Although titanium plates/screws are effective fixation methods (FM) after L-shaped osteotomy reduction malarplasty (LORM), the ideal FM remains controversial. This first finite element analysis (FEA) aimed to study the effect of various zygomatic body/zygomatic arch FM combinations and their placement vectors on the zygoma complex stability after virtual LORM under the effect of both average (150 N/mm2) and maximum (750 N/mm2) forces and three-dimensional (3D) mapping of stress and strain parameters distribution over the zygomatic bone, fixation methods, and total model. RESULTS: The fixation methods about the short-arm of the L-shaped osteotomy showed lower stress, strain, and displacement values than those across the long-arm osteotomy site. Combined with any zygomatic arch fixation methods (ZAFm), the two bicortical screws group (2LS) on the zygomatic body osteotomy site resulted in smaller displacements and the lowest zygoma bone stress and displacement when combined with Mortice-Tenon structure (MT) as zygomatic arch fixation method. Applied forces caused statistically significant differences in zygomatic bone stress (P < 0.001 and P = 0.001) and displacement (P = 0.001 and P = 0.002). CONCLUSION: All FMs both on the zygomatic body and zygomatic arch provide adequate zygomatic complex stability after LORM. The 2LS group showed better resistance than rectangular plate (RP) and square plate (SP) with lower stress concentrations. The L-shaped plate with short-wing on the maxilla (LPwM) is more stable than having the short-wing on the zygoma bone (LPwZ). Future prospective clinical studies are required to validate the current findings.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods.

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Safety of MRI in patients with retained cardiac leads.

PURPOSE: To evaluate the safety of MRI in patients with fragmented retained leads (FRLs) through numerical simulation and phantom experiments. METHODS: Electromagnetic and thermal simulations were performed to determine the worst-case RF heating of 10 patient-derived FRL models during MRI at 1.5 T and 3 T and at imaging landmarks corresponding to head, chest, and abdomen. RF heating measurements were performed in phantoms implanted with reconstructed FRL models that produced highest heating in numerical simulations. The potential for unintended tissue stimulation was assessed through a conservative estimation of the electric field induced in the tissue due to gradient-induced voltages developed along the length of FRLs. RESULTS: In simulations under conservative approach, RF exposure at B1+ ≤ 2 µT generated cumulative equivalent minutes (CEM)43 < 40 at all imaging landmarks at both 1.5 T and 3 T, indicating no thermal damage for acquisition times (TAs) < 10 min. In experiments, the maximum temperature rise when FRLs were positioned at the location of maximum electric field exposure was measured to be 2.4°C at 3 T and 2.1°C at 1.5 T. Electric fields induced in the tissue due to gradient-induced voltages remained below the threshold for cardiac tissue stimulation in all cases. CONCLUSIONS: Simulation and experimental results indicate that patients with FRLs can be scanned safely at both 1.5 T and 3 T with most clinical pulse sequences.

Orientation and depth dependent mechanical properties of the porcine cornea: Experiments and parameter identification.

The porcine cornea is a standard animal model in ophthalmic research, making its biomechanical characterization and modeling important to develop novel treatments such as crosslinking and refractive surgeries. In this study, we present a numerical model of the porcine cornea based on experimental measurements that captures both the depth dependence and orientation dependence of the mechanical response. The mechanical parameters of the established anisotropic hyperelastic material models of Gasser, Holzapfel and Ogden (HGO) and Markert were determined using tensile tests. Corneas were cut with a femtosecond laser in the anterior (100 µm), central (350 µm), and posterior (600 µm) regions into nasal-temporal, superior-inferior, and diagonal strips of 150 µm thickness. These uniformly thick strips were tested at a low speed using a single-axis testing machine. The results showed that the corneal mechanical properties remained constant in the anterior half of the cornea regardless of orientation, but that the material softened in the posterior layer. These results are consistent with the circular orientation of collagen observed in porcine corneas using X-ray scattering. In addition, the parameters obtained for the HGO model were able to reproduce the published inflation tests, indicating that it is suitable for simulating the mechanical response of the entire cornea. Such a model constitutes the basis for in silico platforms to develop new ophthalmic treatments. In this way, researchers can match their experimental surrogate porcine model with a numerical counterpart and validate the prediction of their algorithms in a complete and accessible environment.

Upscaling between an agent-based model (smoothed particle approach) and a continuum-based model for skin contractions.

Skin contraction is an important biophysical process that takes place during and after recovery of deep tissue injury. This process is mainly caused by fibroblasts (skin cells) and myofibroblasts (differentiated fibroblasts which exert larger pulling forces and produce larger amounts of collagen) that both exert pulling forces on the surrounding extracellular matrix (ECM). Modelling is done in multiple scales: agent-based modelling on the microscale and continuum-based modelling on the macroscale. In this manuscript we present some results from our study of the connection between these scales. For the one-dimensional case, we managed to rigorously establish the link between the two modelling approaches for both closed-form solutions and finite-element approximations. For the multi-dimensional case, we computationally evidence the connection between the agent-based and continuum-based modelling approaches.

Comparison of the Design of 3-Pole BLDC Actuators/Motors with a Rotor Based on a Single Permanent Magnet.

Permanent Magnet (PM) Brushless Direct Current (BLDC) actuators/motors have many advantages over conventional machines, including high efficiency, easy controllability over a wide range of operating speeds, etc. There are many prototypes for such motors; some of them have a very complicated construction, and this ensures their high efficiency. However, in the case of household appliances, the most important thing is simplicity, and, thus, the lowest price of the design and production. This article presents a comparison of computer models of different design solutions for a small PM BLDC motor that uses a rotor in the form of a single ferrite magnet. The analyses were performed by using the finite element method. This paper presents unique self-defined parts of basic PM BLDC actuators. With their help, various design solutions were compared with the PM BLDC motor used in household appliances. The authors proved that the reference device is the lightest one and has a lower cogging torque compared to other actuators, but also has a slightly lower driving torque.

Simulation of Rapid Thermal Cycle for Ultra-Fast PCR.

The polymerase chain reaction (PCR) technology is a mainstream detection method used in medical diagnoses, environmental monitoring, food hygiene, and safety. However, the systematic analysis of a compact structure with fast temperature changes for an ultra-fast PCR device that is convenient for on-site detection still lacks investigation. To overcome the problems of low heating efficiency and non-portability of PCR devices currently used, a miniaturized PCR system based on a microfluidic chip, i.e., lab-on-chip technology, has been proposed. The main objective of this paper is to explore the feasibility of using a heat resistor that can reach a fast heating rate and temperature uniformity combined with air cooling technology for rapid cooling and to investigate the influences of various pattern designs and thicknesses of the resistor on heating rates and temperature uniformity. Additionally, a PCR chip made of various materials with different thermal properties, such as surface emissivity, thermal conductivity, mass density, and heat capacity at constant pressure is analyzed. In addition to the heat loss caused by the natural convection of air, the radiation loss of the simulation object is also considered, which makes the model much closer to the practical situation. Our research results provide a considerable reference for the design of the heating and cooling modules used in the ultra-fast PCR protocol, which has great potential in In Vitro Diagnosis (IVD) and the PCR detection of foodborne pathogens and bacteria.

An accurate, robust, and efficient finite element framework with applications to anisotropic, nearly and fully incompressible elasticity.

Fiber-reinforced soft biological tissues are typically modeled as hyperelastic, anisotropic, and nearly incompressible materials. To enforce incompressibility a multiplicative split of the deformation gradient into a volumetric and an isochoric part is a very common approach. However, the finite element analysis of such problems often suffers from severe volumetric locking effects and numerical instabilities. In this paper, we present novel methods to overcome volumetric locking phenomena for using stabilized P1-P1 elements. We introduce different stabilization techniques and demonstrate the high robustness and computational efficiency of the chosen methods. In two benchmark problems from the literature as well as an advanced application to cardiac electromechanics, we compare the approach to standard linear elements and show the accuracy and versatility of the methods to simulate anisotropic, nearly and fully incompressible materials. We demonstrate the potential of this numerical framework to accelerate accurate simulations of biological tissues to the extent of enabling patient-specific parameterization studies, where numerous forward simulations are required.

Vitrifying multiple embryos in different arrangements does not alter the cooling rate.

Vitrification is the most common method of cryopreservation of gametes in fertility clinics due to its improved survival rates compared to slow freezing techniques. For the Open Cryotop® vitrification device, the number of oocytes, or embryos, mounted onto a single device can vary. In this work, a mathematical model is developed for the cooling of oocytes and embryos (samples). The model is solved computationally, to investigate whether varying the number of samples mounted onto the Open Cryotop® affects the cooling rates, and consequently the survival rates, of vitrified samples. Several realistic spatial arrangements of samples are examined, determining their temperature over time. In this way we quantify the effect of spatial arrangement on the cooling rate. Our results indicate that neither the spatial arrangement nor the number of mounted samples has a large effect on cooling rates, so long as the volume of the cryoprotectant remains minimal. The time taken for cooling is found to be on the order of half a second, or less, regardless of the spatial arrangement or number of mounted samples. Hence, rapid cooling can be achieved for any number or arrangement of samples, as long as device manufacturer guidelines are adhered to.

Localization of Deep Ischemia and Hemorrhage in Preterm Infants' Head with Near-Infrared Optical Tomography: A Numerical Case Study.

BACKGROUND AND AIM: Preterm infants have a high incidence of brain lesions that may lead to long-term disabilities. Early diagnosis of cerebral ischemia and hemorrhage may enable protection of the brain by prevention or neuroprotective treatment. Our recently developed time-domain near-infrared optical tomography (TD NIROT) system provides images to diagnose neonatal brain injury. Our aim is to study the image quality achievable from the TD NIROT signals perturbed by noise for two common cases: ischemia and hemorrhage. METHODS: We implemented simulations on a spherical model of diameter 60 mm representing a typical neonatal head where the absorption µa = 0.08 cm-1 and the reduced scattering µ's = 4.1 cm-1. Injury-mimicking spherical inclusions of various diameters (1 ~ 10 mm) were placed at depths of 10 ~ 20 mm in the ischemia case (2.5 × µa) and 14 ~ 30 mm for the hemorrhage case (50 × µa). TD data were generated from a large number of source-detector pairs, i.e., 208 detectors placed within a circle of diameter 40 mm on the surface surrounded by 18 sources. Up to 5% Gaussian noise was added in the simulations. 3D images were reconstructed with the modified Tikhonov minimization with the initial guess of a homogeneous phantom, and the images were evaluated by positional error and Dice similarity. RESULTS: The inclusions were localized correctly with low positional errors (<1 mm), and the segmented images share a high Dice similarity with the ground truth for both the ischemia and the hemorrhage case, even for tiny inclusions of 1 mm in deep tissue. The hemorrhage case with a high contrast tolerates a substantial level of noise even though the performance drops with higher noise as expected. CONCLUSIONS: The large amount of data provided by our novel TD NIROT system provides rich enough information for correctly locating hemorrhage and ischemia in the neonatal brain.

A partition of unity approach to fluid mechanics and fluid-structure interaction.

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

Mechanical Behaviors Research and the Structural Design of a Bipolar Electrostatic Actuation Microbeam Resonator.

A class of bipolar electrostatically actuated micro-resonators is presented in this paper. Two parametric equations are proposed for changing the microbeam shape of the upper and lower sections. The mechanical properties of a micro-resonator can be enhanced by optimizing the two section parameters. The electrostatic force nonlinearity, neutral surface tension, and neutral surface bending are considered in the model. First, the theoretical results are verified with finite element results from COMSOL Multiphysics simulations. The influence of section variation on the electrostatic force, pull-in behaviors and safe working area of the micro-resonator are studied. Moreover, the impact of residual stress on pull-in voltage is discussed. The multi-scale method (MMS) is used to further study the vibration of the microbeam near equilibrium, and the relationship between the two section parameters of the microbeam under linear vibration was determined. The vibration amplitude and resonance frequency are investigated when the two section parameters satisfy the linear vibration. In order to research dynamic analysis under the case of large amplitude. The Simulink dynamics simulation was used to study the influence of section variation on the response frequency. It is found that electrostatic softening increases as the vibration amplitude increases. If the nonlinearity initially shows hardening behavior, the frequency response will shift from hardening to softening as the amplitude increases. The position of softening-hardening transition point decreases with the increase of residual stress. The relationship between DC voltage, section parameters, and softening-hardening transition points is presented. The accuracy of the results is verified using theoretical, numerical, and finite element methods.

A High-Precision Method for Dynamically Measuring Train Wheel Diameter Using Three Laser Displacement Transducers.

Wheel diameter is a significant geometric parameter related to the safe operation of trains, and needs to be measured dynamically. To the best of the authors' knowledge, most existing dynamic measurement methods and systems do not meet the requirement that the wheel diameter measurement error for the high-speed vehicle is less than 0.3 mm. In this paper, a simple method for dynamically and precisely measuring train wheel diameter using three one-dimensional laser displacement transducers (1D-LDTs) is proposed for the first time, and a corresponding measurement system which was developed is described. The factors that affect the measurement accuracy were analyzed. As a main factor, rail deformation caused by the wheel-rail interaction force at low (20 km/h) and high (300 km/h) speeds was determined based on the combination of multi-body dynamics and finite element methods, and the effect of rail deformation on measurement accuracy is greatly reduced by a comparative measurement. Field experiments were performed to verify the performance of the developed measurement system, and the results of the repeatability error and measurement error of the system were both less than 0.3 mm, which meets the requirement of wheel diameter measurements for high-speed vehicles.

The role of spatial variations of abiotic factors in mediating intratumour phenotypic heterogeneity.

We present here a space- and phenotype-structured model of selection dynamics between cancer cells within a solid tumour. In the framework of this model, we combine formal analyses with numerical simulations to investigate in silico the role played by the spatial distribution of abiotic components of the tumour microenvironment in mediating phenotypic selection of cancer cells. Numerical simulations are performed both on the 3D geometry of an in silico multicellular tumour spheroid and on the 3D geometry of an in vivo human hepatic tumour, which was imaged using computerised tomography. The results obtained show that inhomogeneities in the spatial distribution of oxygen, currently observed in solid tumours, can promote the creation of distinct local niches and lead to the selection of different phenotypic variants within the same tumour. This process fosters the emergence of stable phenotypic heterogeneity and supports the presence of hypoxic cells resistant to cytotoxic therapy prior to treatment. Our theoretical results demonstrate the importance of integrating spatial data with ecological principles when evaluating the therapeutic response of solid tumours to cytotoxic therapy.

Finite Element Model for Hydrocephalus and Idiopathic Intracranial Hypertension.

Hydrocephalus and idiopathic intracranial hypertension (IIH) are neuropathies associated with disturbed cerebrospinal fluid dynamics. Several finite element (FE) brain models were suggested to simulate the pathological changes in hydrocephalus, but with overly simplified assumptions regarding the properties of the brain parenchyma. This study proposes a two-dimensional FE brain model, capable of simulating both hydrocephalus and IIH by incorporating poro-hyperelasticity of the brain and detailed structural information (i.e., sulci).

Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data.

We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the [Formula: see text]-projection or the Scott-Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott-Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport.

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

A Systematic Review of Modeling and Simulation for Precision Diamond Wire Sawing of Monocrystalline Silicon.

Precision processing of monocrystalline silicon presents significant challenges due to its unique crystal structure and chemical properties. Effective modeling and simulation are essential for advancing the understanding of the manufacturing process, optimizing design, and refining production parameters to enhance product quality and performance. This review provides a comprehensive analysis of the modeling and simulation techniques applied in the precision machining of monocrystalline silicon using diamond wire sawing. Firstly, the principles of mathematical analytical model, molecular dynamics, and finite element methods as they relate to monocrystalline silicon processing are outlined. Subsequently, the review explores how mathematical analytical models address force-related issues in this context. Molecular dynamics simulations provide valuable insights into atomic-scale processes, including subsurface damage and stress distribution. The finite element method is utilized to investigate temperature variations and abrasive wear during wire cutting. Furthermore, similarities, differences, and complementarities among these three modeling approaches are examined. Finally, future directions for applying these models to precision machining of monocrystalline silicon are discussed.