Coupled Partial Differential Equations Research Articles

A Computational Modeling Approach to Investigate the Influence of Hyperthermia on the Tumor Microenvironment.

The biophysical properties of the tumor microenvironment differ substantially from normal tissues. A constellation of features, including decreased vascularity, lack of lymphatic drainage, and elevated interstitial pressure, diminishes the penetration of therapeutics into tumors. Localhyperthermia within the tumor can alter microenvironmental properties, such as interstitial fluid pressure, potentially leading to improvements in drug penetration. In this context, multi-physics computational models can provide insight into the interplay between the biophysical parameters within the tumor microenvironment and can guide the design and interpretation of experiments that test the bioeffects of local hyperthermia. This paper describes a step-by-step workflow for a computational model coupling partial differential equations describing electrical current distribution, bioheat transfer, and fluid dynamics. The main objective is to study the effects of hyperthermia delivered by a bipolar radiofrequency device on the interstitial fluid pressure within the tumor.The system of mathematical expressions linking electrical current distribution, bioheat transfer, and interstitial fluid pressure is presented, emphasizing the changes in the distribution of the interstitial fluid pressure that could be induced by the thermal intervention.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Assessment of unsteady-state thermal expansion in saccular aneurysms with Newtonian blood circulation including silver nanoparticles

The current study is based on an examination of the effect of Newtonian blood flow on the value of the wall shearing stress of an intracranial aneurysm using computational fluid dynamics. Blood is frequently thought to behave like a Newtonian fluid in cerebral arteries. It is fully described how a saccular aneurysm affects a model of blood-silver (Ag) nanofluid flow via an artery. Blood is regarded as a based fluid because of its rheology. The blood flow is Newtonian, laminar, incompressible, viscous, and unsteady, blood enters with a velocity of 0.08 ms−1 and the arterial wall should be flexible. The novelty of the research is to examine the impact of nanoparticles (Ag) on saccular aneurysm. To find the solution, a time-dependent direct solver was employed. To get the answer, the finite element discretization method (FDM) is applied. Coupled partial differential equations are used to mathematically describe the rheology of blood. In the current analysis, the solutions for velocity, temperature, and pressure distribution are found. Figures are used to illustrate and quickly discuss the effects of physical parameters on the characteristics of arterial blood flow for silver nanoparticles. These findings will be extremely beneficial in the treatment of arteries with aneurysms.

Machine Learning for Semiconductor Process Simulation Described by Coupled Partial Differential Equations

AbstractTechnology computer‐aided design (TCAD) simulation is an important tool for the development of semiconductor devices. Based on coupled partial differential equations (PDEs) for behaviors, TCAD can calculate objects such as impurities, point defects, and electronic carriers in semiconductors. However, over recent years semiconductor devices have become increasingly miniaturized and complicated, resulting in much longer calculation times for TCAD. Machine learning is one technology that may be used to overcome this simulation cost problem. In this study, a neural network architecture is proposed that considers the structure of the coupled PDEs. Features representing each concentration distribution of the calculation objects are extracted by convolution operations and their reaction is modeled by channel attention. The performance of the proposed architecture and of conventional neural network models is evaluated using a simulation dataset generated by 1D coupled PDEs that models the diffusion and reaction of vacancies and interstitial atoms. In addition, the advantage of the method is discussed through the analyses of error correlations of the two predictions and attention coefficients. The machine learning method developed in this study will be applicable to other physics described by coupled PDEs and is expected to speed up the computation of simulations in various fields.

Closed-form solution of oscillating Maxwell nano-fluid with heat and mass transfer

The primary goal of this article is to analyze the oscillating behavior of Maxwell Nano-fluid with regard to heat and mass transfer. Due to high thermal conductivity of engine oil is taken as a base fluid and graphene Nano-particles are introduced in it. Coupled partial differential equations are used to model the governing equations. To evaluate the given differential equations, certain dimensionless factors and Laplace transformations are used. The analytical solution is obtained for temperature, concentration and velocity. The temperature and concentration gradient are also finds to analyze the rate of heat and mass transfer. As a special case, the solution for Newtonian fluid is discussed. Finally, the behaviors of various physical factors are studied graphically for both sine and cosine oscillation and give physical meanings to the parameters.

On the Analytical Treatment for the Fractional-Order Coupled Partial Differential Equations via Fixed Point Formulation and Generalized Fractional Derivative Operators

High-dimensional fractional equation investigation is a cutting-edge discipline with considerable pragmatic and speculative consequences in engineering, epidemiology, and other scientific disciplines. In this study, a hybrid Jafari transform mixed with the Adomian decomposition method for obtaining the analytical solution to Burgers’ problem is provided. Burgers’ equation is a vital mathematical expression that appears in a variety of computational modelling fields, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. By considering a hybrid transform, semianalytical techniques are constructed for the Caputo and Atangana-Baleanu fractional derivative operators. Besides that, existence and uniqueness analyses are carried out with the aid of the Banach contraction-fixed point theory. To obtain the models’ findings, we employed the Jafari transform on fractional-order Burger equations (BEs), supplemented by the inverse Jafari transform. The projected findings for the fractional BEs have been depicted visually. Ultimately, numerical figures are provided to validate the practicality and efficacy. The solution obtained by employing the supplied methodologies has been validated to have the appropriate rate of convergence to the precise solution. The main advantage of the suggested method is the relatively small number of computations performed. It can also be used to address fractional-order scientific issues in a multitude of fields.

Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo gH-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of gH-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and ε-approximate solution of the coupled system. Finally, we propose some work for future research.

Fracture analysis of single edge notched specimen using phase field approach

For the damage tolerant design of cracked structural components, two important parameters namely, residual strength and remaining life are very important. To evaluate these, a reliable fracture parameter such as stress intensity factor (SIF) is to be determined. Phase field (PF) based finite element formulation for a brittle fracture to determine crack propagation trajectory is presented in this paper. PF methodologies have been formulated by coupling partial differential equations (PDEs) after minimizing the total potential energy, which includes elastic strain energy and fracture energy, with respect to the displacement field and PF variable. Solutions of these PDEs for PF variable resulted in identifying crack path geometry, while PDEs displacement field obtained near the crack field is used to determine SIF. In the present study, extending PF formulation to fracture analysis of a structural component is first of its kind. To simulate PF phenomena, a four noded 2-D bilinear quadrilateral element with three degrees of freedom (DOF) at each node (two translations and one for phase field) has been developed to analyze single edge notched (SEN) specimen under tensile and shear loading. By combining the displacement results of PF method with displacement extrapolation technique, SIF has been determined for various crack lengths representing crack propagation in SEN specimens. It is then compared with the analytical SIF solution available in Handbook, and also with the contour integral method SIF results. It is found that the SIF obtained by PF method is in good agreement (±10%) with the other solutions.

Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D)...

Distributed-parameter modeling and dynamic analysis of rotational compressive-mode energy harvesters

This paper presents a modeling and dynamic analysis of a rotational high-efficiency compressive-mode piezoelectric energy harvester (HC-PEH) with partially thickened bow-shaped beams. Based on the Euler–Bernoulli beam theory and the extended Hamilton’s principle, the governing equations of the rotational HC-PEH system are formulated. The nonlinear electromechanical coupling partial differential equations of the system are transformed into ordinary differential equations in the truncated modal coordinates. The developed distributed-parameter model is validated against experimental data, and a good agreement is achieved. The stability and the nonlinear dynamic behavior of the rotational HC-PEH system in conditions of different offset distances and preloaded axial forces are investigated by numerical simulation results. A parametric study of the parameters directly related to the system design is also performed to provide fundamental guidance for understanding the electrical output performance and modulating the voltage–rotation speed responses of the harvester. The result shows that the design parameters of the bow-shaped beam and the PZT plate have apparent effects on the electrical output of the harvester system.

Discussion Paper: A New Mathematical Framework for Representation and Analysis of Coupled PDEs

Abstract We present a computational framework for stability analysis of systems of coupled linear Partial-Differential Equations (PDEs). The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichlet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable. We exploit a new concept of state for PDE systems which allows us to include the boundary conditions directly in the dynamics of the PDE. The resulting algorithms are implemented in Matlab, tested on several motivating and illustrative examples, and the codes have been posted online. Numerical testing indicates the approach has little or no conservatism for a large class of systems and can analyze systems of up to 20 coupled PDEs.

Non-Fourier heat conduction/convection in moving medium

Non-Fourier one-dimensional unsteady heat conduction in a moving medium is investigated by using the Cattaneo-Vernotte- Christov-Jordan (CVCJ) heat flux model for medium speeds less than (sub-critical), equal to (critical), and greater than (super-critical) the thermal wave speed. Coupled partial differential equations are solved simultaneously by a finite volume numerical method. Temperature and heat flux distributions for sub-critical, critical, and super-critical flow conditions are presented for two example problems. The importance of boundary conditions on the thermal wave propagation in both sub-critical and super-critical cases is discussed. Approximate analytical solutions are presented which qualitatively substantiate the numerical results.

Thermoelastic Analysis for Two Collinear Cracks in an Orthotropic Solid Disturbed by Antisymmetrical Linear Heat Flow

The problem of two collinear cracks in an orthotropic solid under antisymmetrical linear heat flow is investigated. It is assumed that there exists thermal resistance to heat conduction through the crack region. Applying the Fourier transform, the thermal coupling partial differential equations are transformed to dual integral equations and then to singular integral equations. The crack-tip thermoelastic fields including the jumps of temperature and elastic displacements on the cracks and the mode II stress intensity factors are obtained explicitly. Numerical results show the effects of the geometries of the cracks and the dimensionless thermal resistance on the temperature change and the mode II stress intensity factors. Also, FEM solutions for the stress intensity factorKare used to compare with the solutions obtained using the method. It is revealed that the friction in closed crack surface region should be considered in analyzing the stress intensity factorK.

Fem Formulation of Coupled Partial Differential Equations for Heat Transfer

Heat Transfer in any field plays an important role for transfer of energy from one region to another region. The heat transfer in porous medium can be simulated with the help of two partial differential equations. These equations need an alternate and relatively easy method due to complexity of the phenomenon involved. This article is dedicated to discuss the finite element formulation of heat transfer in porous medium in Cartesian coordinates. A triangular element is considered to discretize the governing partial differential equations and matrix equations are developed for 3 nodes of element. Iterative approach is used for the two sets of matrix equations involved representing two partial differential equations.

Application of the Local Fractional Variational Iteration Method to Solve System of Coupled Partial Differential Equations Involving Local Fractional Operator

Application of the Local Fractional Variational Iteration Method to Solve System of Coupled Partial Differential Equations Involving Local Fractional Operator

G-JITTER EFFECTS ON THE MIXED CONVECTION FLOW OF NANOFLUID PAST AN INCLINED STRETCHING SHEET

Mixed convection flows of nanofluid past an inclined stretching sheet with g-jitter is studied in this paper. Water based nanofluid containing copper, copper oxide, aluminium oxide and silver nanoparticles are concerned. Coupled nonlinear partial differential equations are solved numerically using an implicit finite difference scheme. The effect of solid nanoparticles volume fraction parameter, amplitude of modulation, frequency of oscillation, mixed convection and inclination angle parameter on the fluid flow and heat transfer characteristics are graphically presented. The reduced skin friction and heat transfer coefficient are decreased when inclination angle and solid nanoparticles volume fraction parameters are increased.

Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory

Transverse dynamical behaviors of axially moving nanoplates which could be used to model the graphene nanosheets or other plate-like nanostructures with axial motion are examined based on the nonlocal elasticity theory. The Hamilton's principle is employed to derive the multivariable coupling partial differential equations governing the transverse motion of the axially moving nanoplates. Subsequently, the equations are transformed into a set of ordinary differential equations by the method of separation of variables. The effects of dimensionless small-scale parameter, axial speed and boundary conditions on the natural frequencies in sub-critical region are discussed by the method of complex mode. Then the Galerkin method is employed to analyze the effects of small-scale parameter on divergent instability and coupled-mode flutter in super-critical region. It is shown that the existence of small-scale parameter contributes to strengthen the stability in the super-critical region, but the stability of the sub-critical region is weakened. The regions of divergent instability and coupled-mode flutter decrease even disappear with an increase in the small-scale parameter. The natural frequencies in sub-critical region show different tendencies with different boundary effects, while the natural frequencies in super-critical region keep constants with the increase of axial speed.

Genetic algorithm based approach to study aligned magnetic field with unsteady forced convection

An approach based on the genetic algorithm is proposed to solve system of coupled differential equation with prescribed boundary conditions. Unsteady, viscous, 2-dimensional incompressible flow over a stretching sheet with outer velocity and aligned magnetic field is considered to illustrate the approach. Coupled partial differential equations are converted to equivalent ordinary differential equation using similarity transform. Results show that this approach may replace traditional shooting technique in solving complex boundary value problems. Comparison with existent methods have been made to show validation of the approach. Graphs and tables are depicted to show impact of aligned magnetic field, stretching parameter on flow and heat transfer over stretching surfaces.

The role of necrosis, acute hypoxia and chronic hypoxia in 18F-FMISO PET image contrast: a computational modelling study

Positron emission tomography (PET) using 18F-fluoromisonidazole (FMISO) is a promising technique for imaging tumour hypoxia, and a potential target for radiotherapy dose-painting. However, the relationship between FMISO uptake and oxygen partial pressure () is yet to be quantified fully. Tissue oxygenation varies over distances much smaller than clinical PET resolution (<100 μm versus ∼4 mm), and cyclic variations in tumour perfusion have been observed on timescales shorter than typical FMISO PET studies (∼20 min versus a few hours). Furthermore, tracer uptake may be decreased in voxels containing some degree of necrosis.This work develops a computational model of FMISO uptake in millimetre-scale tumour regions. Coupled partial differential equations govern the evolution of oxygen and FMISO distributions, and a dynamic vascular source map represents temporal variations in perfusion. Local FMISO binding capacity is modulated by the necrotic fraction. Outputs include spatiotemporal maps of and tracer accumulation, enabling calculation of tissue-to-blood ratios (TBRs) and time-activity curves (TACs) as a function of mean tissue oxygenation.The model is characterised using experimental data, finding half-maximal FMISO binding at local of 1.4 mmHg (95% CI: 0.3–2.6 mmHg) and half-maximal necrosis at 1.2 mmHg (0.1–4.9 mmHg). Simulations predict a non-linear non-monotonic relationship between FMISO activity (4 hr post-injection) and mean tissue : tracer uptake rises sharply from negligible levels in avascular tissue, peaking at ∼5 mmHg and declining towards blood activity in well-oxygenated conditions. Greater temporal variation in perfusion increases peak TBRs (range 2.20–5.27) as a result of smaller predicted necrotic fraction, rather than fundamental differences in FMISO accumulation under acute hypoxia. Identical late FMISO uptake can occur in regions with differing and necrotic fraction, but simulated TACs indicate that additional early-phase information may allow discrimination of hypoxic and necrotic signals.We conclude that a robust approach to FMISO interpretation (and dose-painting prescription) is likely to be based on dynamic PET analysis.

메탄 하이드레이트 생산 묘사를 위한 수치도구의 개발

방대한 저장량으로 차세대 에너지원으로 평가받는 메탄가스 하이드레이트는 생산과정에서 유발될 수 있는 문제를 최소화하고 최적의 생산조건을 선정하기 위한 하이드레이트 포함한 다공질 재료의 THM 현상에 대한 프로그램의 개발이 절실하다. 기존의 해석 프로그램들은 국제공동연구를 통하여 프로그램들간의 상호 비교검증을 진행하고 있으나, 예측값의 불일치와 수렴성에 문제가 있는 것으로 나타났다. 본 논문에서는 다공질 재료내 메탄 하이드레이트의 해리 현상을 해석할 수 있는 fully coupled THM 유한요소 프로그램을 개발하였다. Methane hydrate, soil, water, 및 methane gas의 질량보존의 법칙, 에너지 보존의 법칙, 그리고 힘평형 방정식으로부터 지배방정식을 유도하였다. 다양한 주변수들의 조합을 통하여 주변수를 변위, 가스 포화도, 유체압, 온도, 하이드레이트 포화도로 선택하였으며, 상변화 전영역에서 해석이 가능하도록 하였다. 하이드레이트의 해리를 예측하는 모델은 kinetic model을 이용하였다. 개발된 THM 유한요소 프로그램을 이용하여 메탄가스 생산에 관한 Masuda의 실내 모형실험 결과와 비교적 분석을 수행하였으며, 해의 수렴성과 안정성을 확인할 수 있었다. Methane gas hydrate which is considered energy source for the next generation has an urgent need to develop reliable numerical simulator for coupled THM phenomena in the porous media, to minimize problems arising during the production and optimize production procedures. International collaborations to improve previous numerical codes are in progress, but they still have mismatch in the predicted value and unstable convergence. In this paper, FEM code for fully coupled THM phenomena is developed to analyze methane hydrate dissociation in the porous media. Coupled partial differential equations are derived from four mass balance equations (methane hydrate, soil, water, and hydrate gas), energy balance equation, and force equilibrium equation. Five main variables (displacement, gas saturation, fluid pressure, temperature, and hydrate saturation) are chosen to give higher numerical convergence through trial combinations of variables, and they can analyze the whole region of a phase change in hydrate bearing porous media. The kinetic model is used to predict dissociation of methane hydrate. Developed THM FEM code is applied to the comparative study on a Masuda's laboratory experiment for the hydrate production, and verified for the stability and convergence.