Numerical Techniques Research Articles

Biomechanical changes in the proximal femur before and after removal of femoral neck system

BackgroundAs an innovative internal fixation system, FNS (femoral neck system) is increasingly being utilized by surgeons for the treatment of femoral neck fractures. At present, there have been numerous finite element analysis experiments studying the immediate stability of FNS and CSS in treating femoral neck fractures. However, there is scarce mechanical analysis available regarding the effects post internal fixation removal. This study aimed to investigate the alterations in mechanical parameters of the proximal femur before and after the removal of FNS (femoral neck system), and to assess potential distinctions in indicators following the extraction of CSS (Cannulated Screws).MethodsA proximal femur model was reconstructed using finite element numerical techniques. The models for CSS and FNS were formulated utilizing characteristics and parametric definitions. The internal fixation was combined with a normal proximal femur model to simulate the healing state after fracture surgery. Within the framework of static analysis, consistent stress burdens were applied across the entirety of the models. The total deformation and equivalent stress of the proximal femur were recorded before and after the removal of internal fixation.ResultsUnder the standing condition, the total deformation of the model before and after removing CSS was 0.99 mm and 1.10 mm, respectively, indicating an increase of 12%. The total deformation of the model before and after removing FNS was 0.65 mm and 0.76 mm, respectively, indicating an increase of 17%. The equivalent stress for CSS and FNS were 55.21 MPa and 250.67 MPa, respectively. The average equivalent stress on the cross-section of the femoral neck before and after removal of CSS was 7.76 MPa and 6.11 MPa, respectively. The average equivalent stress on the cross-section of the femoral neck before and after removal of FNS was 9.89 MPa and 8.79 MPa, respectively.ConclusionsThe retention of internal fixation may contribute to improved stability of the proximal femur. However, there still existed risks of stress concentration in internal fixation and stress shielding in the proximal femur. Compared to CSS, the removal of FNS results in larger bone tunnels and insufficient model stability. Further clinical interventions are recommended to address this issue.

Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials

Abstract The calendering process is pivotal in enhancing various materials’ surface properties and characteristics, making them indispensable for achieving desired product quality and performance. Also, this process holds significant relevance in various industrial applications, such as polymer processing, food production, and the manufacturing of composite materials. So, the aim of this study is to theoretically examine the calendering process applied to third-grade materials. It specifically explores how temperature variations impact material behavior during passage through two counter-rotating heated rolls. Particular consideration is given to the influence of temperature-dependent viscosity via Reynold’s model. The complexities of mass, momentum, and energy balance equations are reduced through the application of the Lubrication approximation theory. Solutions to these equations for variables such as velocity, flow rate, and temperature fields are accomplished by combining perturbation and numerical techniques. In relation to the calendering process, the thickness of the exiting sheet is specifically explored. Furthermore, this study quantifies substantial engineering parameters such as roll-separating force, pressure distribution, and power transferal from the rolls to the fluid. The governing equations belong to three key dimensionless parameters, namely, the Brinkman number, which is a product of Eckert number and Prandtl number, the temperature-dependent consistency index μ \mu , and a parameter η \eta correlating to non-Newtonian behavior. The outcomes of this study are presented both graphically and in tabular form. It has been observed that a rise in the third-grade parameter decreases detachment point and sheet thickness due to increased material rigidity. Furthermore, established results in the literature regarding the calendering of Newtonian fluids are validated.

Photovoltaic spatial gap solitons in biased photorefractive optical lattices

We investigate the existence and characteristics of spatially confined optical gap solitons within an optical lattice embedded in a biased bulk photovoltaic photorefractive crystal. The Floquet-Bloch theory is used to analyze the uniform lattice and derive the optical lattice band structure. In the photonic band gaps, which are typically opaque to light transmission, the photorefractive nonlinearity permits the formation of solitons. The paraxial Helmholtz equation is set up and solved thereby discovering single hump and double hump soliton states in both band gaps. Interestingly, we have not found any multi peak solitons to exist in this particular case. We examine the characteristics of these gap solitons, finding that the soliton width (an indicator of nonlinearity) and intensity depend on their location within the band gap. We find that the magnitude of the external electric field profoundly affects the gap soliton characteristics. Additionally, the Vakhitov-Kolokolov (VK) criterion, perturbation analysis and numerical techniques are used to analyze the stability of the various types of the spatial gap solitons in both the band gaps.

A Novel Consolidation Analysis Framework: Universal Function Approximators Regularized by Physical Principles

Analytical and numerical techniques are widely used to analyse and interpret soil consolidation problems. An important limitation is the requirement for significant geotechnical knowledge and expertise to derive ‘true’ solutions. This study proposes an alternative: a universal function approximator regularized with known physical principles. The proposed approach here advances previous work to solve one-dimensional (1D) consolidation considering both self-weight and large strains, as well as two-dimensional (2D) consolidation by vertical drains. Initial and boundary conditions of the studied consolidation problems are first strongly and weakly enforced for comparison. A neural network is adopted here as the function approximator. To boost prediction accuracy, a novel strategy is proposed to adaptively sample data points for training. An estimate of epistemic uncertainty is achieved using the confidence interval of ensembled multiple outputs. The results show that the proposed approach accurately predicts the behaviours of complex consolidation processes. Results also indicate that regularization using weak physical constraints can alleviate the imbalance of back-propagated gradients of different loss terms and, in turn, achieve higher accuracy. The proposed method is generic, mesh-free, more robust and can be applied to a wide range of geotechnical problems.

A novel hybrid finite element – Finite difference approach for 3D Magnetotelluric modeling

A new method is developed to solve the Magnetotelluric (MT) forward problem in three dimensions (3D). The finite-differences (FD) and the finite-elements (FE) methods are considered to be popular numerical techniques to solve the partial differential equations in electromagnetics. The FD approach is known to be easily applicable to a given problem while the FE method is much more flexible when a topography is present, however, it is also computationally more expensive to solve the linear matrix equations for the FE due to the number of non-zero entries it populates when compared to the FD’s. For that reason, a new algorithm is offered to mix the FE and FD in a single mesh using hexahedra and rectangular prisms in 3D. By doing that, it is possible to take advantage of the beneficial aspects of these two methods. It was found out that the newly developed hybrid approach (HYB) is almost as fast as the FD method while being very memory efficient which is an important aspect due to the limited memory space of a GPU which is employed to solve the systems iteratively. It was found that the HYB method is as accurate as the FE method while it is a more reasonable choice for larger models due to the solution speed of the FE method as it does not scale well with the increased model size. On average, a speed-up of 1.6 is found between the FE and HYB methods with the three models tested and it increases with the increasing model size. Using a GPU during the iterative solution made it possible to solve the linear systems in seconds and with the help of the HYB method, the memory consumption was found to be less than half of what the FE method needed. This ultimately means that much larger models can be simulated on the limited memory that a GPU has. Finally, the technique shown to implement the HYB approach makes the deployment of it to existing FE codes so much easier.

Shape optimization for interface identification in nonlocal models

AbstractShape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface–dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques. The code for obtaining the results in this paper is published at (https://github.com/schustermatthias/nlshape).

Fractional Stochastic Partial Differential Equations: Numerical Advances and Practical Applications—A State of the Art Review

Fractional Stochastic Partial Differential Equations: Numerical Advances and Practical Applications—A State of the Art Review

A Practical probabilistic history matching framework for characterizing fracture network parameters of shale gas reservoirs

Simulating fluid flow in complex fracture systems encountered in subsurface systems, such as those encountered in tight or shale gas reservoirs, presents considerable challenges. Sophisticated numerical techniques and characterization approaches are required to integrate static and dynamic data from multiple sources to account for the intricate interplay between fractures and the surrounding rock matrix. A novel workflow using an indicator-based probability perturbation method (PPM) is applied to characterize uncertain discrete fracture network parameters. The posterior probability distributions of primary and secondary fracture transmissivity, aperture, length, height, and intensity of the secondary fracture are estimated according to the production (flow) histories. A case study based on the Horn River shale gas reservoir is presented. A pilot point scheme is formulated to update the distributions of P32L. The forward modelling entails upscaling each fracture model into a dual-porosity dual-permeability model and performing multiphase flow simulation. This approach produces an ensemble of history-matched discrete fracture and upscaled models by incorporating static and dynamic data. The results demonstrate the utility of the developed method for estimating secondary fracture parameters, which are not inferrable from other static information alone. The inference of both primary and secondary fractures offers a more comprehensive characterization of the fracture networks in tight or shale gas reservoirs. Integrating a pilot point parameterization technique and a sequential simulation approach into the PPM framework presents significant novelties in contributing to a comprehensive and effective method of simulating fluid flow in complex fracture systems, addressing the challenges associated with non-linear relationships between fracture model parameters and the corresponding flow response.

Variance-Reduced Stochastic Optimization for Efficient Inference of Hidden Markov Models

Hidden Markov models (HMMs) are popular models to identify a finite number of latent states from sequential data. However, fitting them to large data sets can be computationally demanding because most likelihood maximization techniques require iterating through the entire underlying data set for every parameter update. We propose a novel optimization algorithm that updates the parameters of an HMM without iterating through the entire data set. Namely, we combine a partial E step with variance-reduced stochastic optimization within the M step. We prove the algorithm converges under certain regularity conditions. We test our algorithm empirically using a simulation study as well as a case study of kinematic data collected using suction-cup attached biologgers from eight northern resident killer whales (Orcinus orca) off the western coast of Canada. In both, our algorithm converges in fewer epochs, with less computation time, and to regions of higher likelihood compared to standard numerical optimization techniques. Our algorithm allows practitioners to fit complicated HMMs to large time-series data sets more efficiently than existing baselines. Supplemental materials are available online.

Troesch’s problem: A numerical study with cubic trigonometric B-spline method

This research paper presents a numerical investigation of Troesch’s problem using the collocation method based on the cubic trigonometric B-Spline technique. Troesch’s problem is a well-known BVP that arises in the field of mathematical physics and engineering. The objective of this study is to provide an accurate and efficient numerical solution to Troesch’s problem. Our method utilizes the properties of cubic trigonometric B-splines with collocation technique to approximate the solution of Troesch’s problem. The method is employed to satisfy the boundary conditions, resulting in a system of algebraic equations. A comprehensive error analysis is conducted to assess the accuracy and convergence of the proposed scheme. The obtained numerical results are compared with the exact solution, where available, and with other existing numerical methods to validate the accuracy and efficiency of the proposed technique.The significance and relevance of this work lie in its ability to provide highly accurate solutions to Troesch’s problem. Furthermore, the proposed method exhibits good convergence properties and demonstrates superiority over other numerical techniques in terms of accuracy and efficiency. These findings have practical implications in the field of mathematical physics and engineering, where precise solutions to Troesch’s problem can greatly enhance the understanding and optimization of various systems.

A comprehensive approach for understanding debris flow interaction with pipelines through dynamic impact pressure modeling

The interaction of debris flows, and impacted structures is still not well understood due to the complex mechanisms of the flows. Although several methodologies from experimental to numerical have developed so far, a thorough assessment of impact mechanism necessitates multiphase modelling approaches. Therefore, this paper presents a dynamic impact pressure modelling approach of debris flows using experimental and numerical techniques. The experimental investigations involved seven type of debris flows based on solid volume fraction (αs), that being released on an inclined flume, impacting a 2″(52 mm) pipe model. The numerical investigations were performed in Computational Fluid Dynamics (CFD) environment by utilising the Spalart-Allmaras turbulence and Hershel- Bulkley rheological models by considering different inclined angle (β) impact scenarios. The findings revealed that all debris flows produced were in supercritical flows regime and characterized as dilute, medium viscous, and highly viscous flows, based on Froude (Fr) and Reynold (Re) numbers. Drag coefficients (CD-90/CD-0) were behaved non-linearly with respect to Fr and Re. The proposed refined dynamic normal (pd-90) and axial (pd-0) impact models using different set of Fr, Re and β will offer valuable insight for pipeline operators seeking to comprehend the complex interaction between debris flows and pipelines.

Spectral neural approximations for models of transcriptional dynamics

The advent of high-throughput transcriptomics provides an opportunity to advance mechanistic understanding of transcriptional processes and their connections to cellular function at an unprecedented, genome-wide scale. These transcriptional systems, which involve discrete stochastic events, are naturally modeled using chemical master equations (CMEs), which can be solved for probability distributions to fit biophysical rates that govern system dynamics. While CME models have been used as standards in fluorescence transcriptomics for decades to analyze single-species RNA distributions, there are often no closed-form solutions to CMEs that model multiple species, such as nascent and mature RNA transcript counts. This has prevented the application of standard likelihood-based statistical methods for analyzing high-throughput, multi-species transcriptomic datasets using biophysical models. Inspired by recent work in machine learning to learn solutions to complex dynamical systems, we leverage neural networks and statistical understanding of system distributions to produce accurate approximations to a steady-state bivariate distribution for a model of the RNA life cycle that includes nascent and mature molecules. The steady-state distribution to this simple model has no closed-form solution and requires intensive numerical solving techniques: our approach reduces likelihood evaluation time by several orders of magnitude. We demonstrate two approaches, whereby solutions are approximated by 1) learning the weights of kernel distributions with constrained parameters or 2) learning both weights and scaling factors for parameters of kernel distributions. We show that our strategies, denoted by kernel weight regression and parameter-scaled kernel weight regression, respectively, enable broad exploration of parameter space and can be used in existing likelihood frameworks to infer transcriptional burst sizes, RNA splicing rates, and mRNA degradation rates from experimental transcriptomic data.

Improving the hepatitis viral transmission model’s dynamics by vaccination and contrasting it with the fractional-order model

We investigate mathematical models of the Hepatitis B and C viruses in the study, considering vaccination effects into account. By utilising fractional and ordinary differential equations, we prove the existence of equilibrium and the well-posedness of the solution. We prove worldwide stability with respect to the fundamental reproduction number. Our numerical techniques highlight the biological relevance and highlight the effect of fractional derivatives on temporal behaviour. We illustrate the relationships among susceptible, immunised, and infected populations in our epidemiological model. Using comprehensive numerical simulations, we analyse the effects of fractional derivatives and highlight solution behaviours. Subsequent investigations will examine the impact of regional heterogeneity, providing significant perspectives for epidemiological research.

The Bredehoeft problem: Evaluating salvage during groundwater pumping in unconfined aquifers

During groundwater pumping from unconfined aquifers, evapotranspiration can be reduced as the water table drops, a process known as salvage. This phenomenon must be included in water balance assessments for accurate predictions of groundwater withdrawals. Although the salvage phenomenon has been well understood for a long time, i.e., since Theis’ pioneering studies, the models that were later developed by Bredehoeft always relied on numerical techniques to treat the inherent non-linearity, most commonly using MODFLOW. Analytical methods have been previously used with success for practical evaluation of water balance components such as stream depletion rates. This work presents a novel analytical solution of the Bredehoeft problem. It is a breakthrough in analytical evaluation of transient salvage losses induced by groundwater withdrawals due to declines of water table. We evaluate stream depletion and loss of storage in addition to salvage, using the new model. Detailed analysis of the solution using COMSOL software indicates that the linear approximation used in the analytical solution is accurate for most practical cases. Diagrams for drawdown, water balance and evaluation of solution accuracy are presented, considering a broad range of problem parameters. Results can be used for analyses of watershed-scale water balances.

Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt [Formula: see text] model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical [Formula: see text] equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding [Formula: see text] model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.

Evaluation of Aortic Valve Pressure Gradients for Increasing Severities of Rheumatic and Calcific Stenosis Using Empirical and Numerical Approaches

The evaluation and accurate diagnosis of the type and severity of aortic stenosis relies on the precision of medical imaging technology and clinical correlations and the expertise of medical professionals. The application of the clinical correlation to different aortic stenosis morphologies and severities is investigated. The manner in which numerical techniques can be used to simulate the blood flow through pathological aortic valves was analysed and compared to the ground-truth CFD model. Larger pressure gradients are estimated in all severities of rheumatic aortic valves compared to calcific aortic valves. The zero-dimensional morphology-insensitive model underpredicted the transvalvular pressure gradient with the greatest error. The 1D model underestimated the pressure gradient in rheumatic cases and overestimated the pressure gradient in calcific cases. The pressure gradients estimated by the clinical approach depends on the location of the flow vena contracta and is sensitive to the severity and type of valve lesion. Through the analysis of entropy generation within the flow domain, the dominant parameters and regions driving adverse pressure gradients were identified. It is concluded that sudden expansion is the dominant parameter leading to higher pressure gradients in rheumatic heart valves compared to calcific ones.

Exploring Data Input Problems in Mixed Reality Environments: Proposal and Evaluation of Natural Interaction Techniques

Data input within mixed reality environments poses significant interaction challenges, notably in immersive visual analytics applications. This study assesses five numerical input techniques: three benchmark methods (Touch-Slider, Keyboard, Pinch-Slider) and two innovative multimodal techniques (Bimanual Scaling, Gesture and Voice). An experimental design was employed to compare these techniques’ input efficiency, accuracy, and user experience across varying precision and distance conditions. The findings reveal that multimodal techniques surpass slider methods in input efficiency yet are comparable to keyboards; the voice method excels in reducing cognitive load but falls short in accuracy; and the scaling method marginally leads in user satisfaction but imposes a higher physical load. Furthermore, this study outlines these techniques’ pros and cons and offers design guidelines and future research directions.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

Thermal performance of mixed convective radiative peristaltic flow of Bingham nanofluid

The significance of peristaltic phenomena has generated substantial interest within the fields of biological and biomedical engineering. Therefore, the present study deals with magnetohydrodynamics (MHD) peristaltic flow of Bingham nanofluid in a channel. Here the channel walls are symmetric and elastic in nature. Velocity slip, heat and mass convective conditions are enforced on channel walls. The aspects of dissipation, Joule heating and mixed convection is analyzed. Thermal transport included the features of thermal radiation, Brownian and thermophoresis motion. Analysis and modelling are investigated in the presence of the phenomenon heat generation and absorption. The analysis has been wrapped up under the assumptions of a long wavelength and a low Reynolds number. Numerical techniques are implemented for finding the solutions of governing differential system. Results for velocity, concentration, thermal field, rate of heat transfer, skin friction coefficient, Nusselt number and Sherwood number are analyzed subject to pertinent parameters. The study reveals that velocity is enhanced via both thermal Grashof number and velocity slip parameter. Temperature shows opposite behavior against radiation and Brinkman number. Rate of heat transfer enhances for larger values Brownian motion and Hartman parameters. This study's practical applications include the use of peristaltic pumps for drug delivery in the pharmaceutical and chemical industries.

Channel-attention-based LSTM network for modeling temperature-induced responses of cable-stayed bridges

Temperature has a significant impact on cable-stayed bridges, yielding structural responses comparable to those from vehicular loads, winds, etc. However, advanced numerical techniques for evaluating long-term temperature-induced responses (TIRs) of cable-stayed bridges are complicated and computationally inefficient. Therefore, this study leverages recent advances in deep learning and develops a channel-attention-based bidirectional long short-term memory network (CABLe) to directly get the complex mapping between structural temperatures and TIRs from the monitoring data. The key concept behind is the proposed channel attention mechanism (CAM), where its attention weights are calculated using a cosine similarity between latent sequential features to find the most informative contents of the signal. A comparison study is conducted with the bidirectional long short-term memory (BiLSTM) to show the benefits of the proposed CAM. The proposed method successfully predicts TIRs of a cable-stayed bridge using the imbalanced data. Results indicate that the CABLe outperforms the BiLSTM network and shows a high prediction accuracy with unseen temperature data.