Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is Newton’s method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this short note, we seek to enlarge the basin of attraction of the classical Newton’s method. The key idea is to develop a relatively simple multiplicative transform of the original equations, which leads to a reduction in nonlinearity, thereby alleviating the limitation of Newton’s method. Based on this idea, we derive a new class of iterative methods and rediscover Halley’s method as the limit case. We present the application of these methods to several mathematical functions (real, complex, and vector equations). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses. For scalar equations, the increase in computational cost per iteration is minimal. For vector functions, more extensive analysis is needed to compare the increase in cost per iteration and the improvement in convergence of specific problem

A Bayesian constitutive model selection framework for biaxial mechanical testing of planar soft tissues: Application to porcine aortic valves

A variety of constitutive models have been developed for soft tissue mechanics. However, there is no established criterion to select a suitable model for a specific application. Although the model that best fits the experimental data can be deemed the most suitable model, this practice often can be insufficient given the inter-sample variability of experimental observations. Herein, we present a Bayesian approach to calculate the relative probabilities of constitutive models based on biaxial mechanical testing of tissue samples. 46 samples of porcine aortic valve tissue were tested using a biaxial stretching setup. For each sample, seven ratios of stresses along and perpendicular to the fiber direction were applied. The probabilities of eight invariant-based constitutive models were calculated based on the experimental data using the proposed model selection framework. The calculated probabilities showed that, out of the considered models and based on the information available through the utilized experimental dataset, the May–Newman model was the most probable model for the porcine aortic valve data. When the samples were grouped into different cusp types, the May–Newman model remained the most probable for the left- and right-coronary cusps, whereas for non-coronary cusps two models were found to be equally probable: the Lee–Sacks model and the May–Newman model. This difference between cusp types was found to be associated with the first principal component analysis (PCA) mode, where this mode’s amplitudes of the non-coronary and right-coronary cusps were found to be significantly different. Our results show that a PCA-based statistical model can capture significant variations in the mechanical properties of soft tissues. The presented framework is applicable to any tissue type, and has the potential to provide a structured and rational way of making simulations population-based

An Information-Theoretic Framework for Optimal Design: Analysis of Protocols for Estimating Soft Tissue Parameters in Biaxial Experiments

A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework considers the information gain about the parameters from the experiment as the key criterion to be maximised, which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have a relatively low information content compared to high angles. The results also show that a smaller number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly D-optimal design

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow

An information-theoretic framework for optimal design: analysis of protocols for estimating soft tissue parameters in biaxial experiments

A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information, and their combination is proposed. The framework is tested on the analysis of protocols-combination of angles along which strain measurements can be acquired-in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework sees information gain about the parameters from the experiment as the key criterion to be maximised which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have relatively low information content compared to high angles. They also show that fewer number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly the D-optimal design

On improving the numerical convergence of highly nonlinear elasticity problems

Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load step sizes. In this work, we present a new method to transform the discretized governing equations so that the transformed problem has significantly reduced nonlinearity and, therefore, Newton solvers exhibit improved convergence properties. We study exponential-type nonlinearity in soft tissues and geometric nonlinearity in compression, and propose novel formulations for the two problems. We test the new formulations in several numerical examples and show significant reduction in iterations required for convergence, especially at large load steps. Notably, the proposed formulation is capable of yielding convergent solution even when 10–100 times larger load steps are applied. The proposed framework is generic and can be applied to other types of nonlinearities as well

Neighborhood physical food environment and cardiovascular risk factors in India: Cross-sectional evidence from APCAPS

There has been increasing interest in associations between neighborhood food environments and cardiovascular risk factors. However, results from high-income countries remain inconsistent, and there has been limited re- search from low- and middle-income countries. We conducted a cross-sectional analysis of the third wave follow- up of the Andhra Pradesh children and parents study (APCAPS) (n = 5764, median age 28.8 years) in south India. We examined associations between the neighborhood availability (vendor density per km2 within 400 m and 1600 m buffers of households) and accessibility (distance from the household to the nearest vendor) of fruit/ vegetable and highly processed/take-away food vendors with 11 cardiovascular risk factors, including adiposity measures, glucose-insulin, blood pressure, and lipid profile. In fully adjusted models, higher density of fruit/ vegetable vendors within 400 m of participant households was associated with lower systolic blood pressure [−0.09 mmHg, 95% confidence interval (CI): −0.17, −0.02] and diastolic blood pressure (−0.10 mmHg, 95% CI: −0.17, −0.04). Higher density of highly processed/take-away food vendors within 400 m of participant households was associated with higher Body Mass Index (0.01 Kg/m2, 95% CI: 0.00, 0.01), waist circumference (0.22 mm, 95% CI: 0.05, 0.39), systolic blood pressure (0.03 mmHg, 95% CI: 0.01, 0.06), and diastolic blood pressure (0.03 mmHg, 95% CI: 0.01, 0.05). However, within 1600 m buffer, only association with blood pressure remained robust. No associations were found for between neighborhood accessibility and cardiovascular risk factors. Lower density of fruit/vegetable vendors, and higher density of highly processed/take-away food ven- dors were associated with adverse cardiovascular risk profiles. Public health policies regarding neighborhood food environments should be encouraged in south India and other rural communities in south Asia

Association of Neighborhood Alcohol Environment With Alcohol Intake and Cardiovascular Risk Factors in India: Cross-Sectional Evidence From APCAPS.

There are more and more proofs about the impact of neighborhood alcohol environment on alcohol-associated events. The relationship between the neighborhood availability and accessibility of alcohol outlet with individual level of alcohol consumption along with 11 cardiovascular risk factors was explored for the first time in India using data from the 3rd follow-up of the Andhra Pradesh children and parents study (APCAPS) (n = 6156, for liquor intake and 5,641 for heart and blood vessel risk elements). In fully adjusted models, volunteers in the lowest tertile performed worse than volunteers in the highest tertile of distance to the closest alcohol outlet were more probably to exhibit less alcohol consumption (-14.40 g/day, 95% CI: -26.21, -2.59). A unit per km2 rise in alcohol outlet density in 400 m buffering area was related to a rise in waist circumference (1.45 mm, 95% CI: 0.13, 2.77), SBP (0.29 mmHg, 95% CI: 0.09, 0.49), and DBP (0.19 mmHg, 95% CI: 0.03, 0.35). A unit per 100 m rise in distance to the closest alcohol outlet was related to a rise in waist circumference (-2.39 mm, 95% CI: -4.18, -0.59), SBP (-0.41 mmHg, 95% CI: -0.68, -0.15), and DBP (-0.29 mmHg, 95% CI: -0.51, -0.07). Neighborhood availability of alcohol outlets within immediate locality of participants' households had a closer relationship with cardiovascular risk factors than that within the whole village. Public health policies designed to limit neighborhood availability and accessibility of alcohol outlets ought to be advocated in southern India