Mathematical modelling and numerical simulation of the morphological development of neurons

Background

The morphological development of neurons is a very complex process involving both genetic and environmental components. Mathematical modelling and numerical simulation are valuable tools in helping us unravel particular aspects of how individual neurons grow their characteristic morphologies and eventually form appropriate networks with each other.

Methods

A variety of mathematical models that consider (1) neurite initiation (2) neurite elongation (3) axon pathfinding, and (4) neurite branching and dendritic shape formation are reviewed. The different mathematical techniques employed are also described.

Results

Some comparison of modelling results with experimental data is made. A critique of different modelling techniques is given, leading to a proposal for a unified modelling environment for models of neuronal development.

Conclusion

A unified mathematical and numerical simulation framework should lead to an expansion of work on models of neuronal development, as has occurred with compartmental models of neuronal electrical activity.

     

High-order numerical simulation and modelling of the interaction of energetic and inert materials

We present an integrated algorithm on a Eulerian grid, for multimaterial simulations of energetic and inert materials modelled by non-ideal equations of state. We employ high-resolution shock capturing numerical algorithms for each material inside its domain and use an overlap domain method across the interface, maintained by a recently developed, hybrid, level-set algorithm. For applications to condensed explosives we implement a non-ideal, wide-ranging equation of state and reaction rate law. For inert materials, like plastic, metal, water, etc., we implement a (linear in the pressure) Mie–Grüneisen, (U _p ?U _s ), equation of state. We present a series of verifications of the integrated multimaterial code and show validations against experiment. We show examples of simulations of various experiments associated with real or planned experiments, some of which contain energetic materials (specifically the condensed explosives PBX-9502 and PBX-9501).     

Limitations of mathematical modelling and numerical simulation of industrial and laboratory high-pressure processes

High pressures up to several hundreds of MPa are utilised in a wide range of applications in chemical engineering, bioengineering, and food engineering, aiming at selective control of (bio-)chemical reactions. Non-uniformity of process conditions may threaten the safety and quality of the resulting products as the process conditions such as pressure, temperature, and treatment history are crucial for the course of (bio-)chemical reactions. Therefore, thermofluid dynamical phenomena during the high-pressure process have to be examined, and tools to predict process uniformity and to optimise the processes have to be developed. Recently, mathematical models and numerical simulations of laboratory and industrial scale high-pressure processes have been set up and validated by experimental results. This contribution deals with the assumption of the modelling that relevant (bio-)chemical compounds are ideally dissolved or diluted particles in a continuum flow. By considering the definition of the continuum hypothesis regarding the minimum particle population in a distinct volume, limitations of this modelling and simulation are addressed.     

Mathematical modelling of hysteresis

Summary This paper presents a formal mathematical treatment of the general properties of scalar hysteretic systems. A generalization of the Preisach representation including both irreversible and reversible components of magnetization is given. It is shown that the Mayergoyz's theorem has a more general validity in this new representation. The effect of a linear feedback transformation on the existence of the generalized Preisach representation is considered, and the necessary and sufficient conditions for the invariance of such an existence are given. Finally the effect of the shape demagnetizing field on the relative weight of reversible and irreversible components of magnetization is discussed. The reversible component is found to be enhanced by the shape demagnetizing field. The results obtained in this paper are relevant not only for magnetic systems, but also for the mathematical modelling of any kind of hysteretic system.

---

Riassunto In questo lavoro si presenta una trattazione matematica formale delle proprietà generali dei sistemi isteretici scalari. Si dà una generalizzazione del modello di Preisach che include entrambe le componenti irreversibile e reversibile della magnetizzazione. Si mostra come il teorema di Mayergoyz abbia una piú generale validità nella nuova rappresentazione. Si considera l'effetto di una trasformazione lineare di retroazione sull'esistenza del modello di Preisach generalizzato, e si danno le condizioni necessarie e sufficienti per l'invarianza di tale esistenza. Si esamina infine l'effetto del campo smagnetizzante sul peso relativo delle componenti reversibili e irreversibili della magnetizzazione. Si trova che il peso della componente reversibile cresce al crescere del fattore di smagnetizzazione. I risultati ottenuti sono di interesse non solo per i sistemi magnetici, ma anche per la modellizzazione matematica di qualsiasi sistema isteretico.

---

Резюме В этой статье предлагается формалвное математическое рассмотрение общих свойств скалярных гистерезисных систем. Приводится обобщение представления Прейсака, включая необратимые и обратимые компоненты намагничивания. Показывается, что теория Майергойза имеет более широкую применимость в этом новом представлении. Приводятся необходимые и достаточные условия для инвариантности существования обобщенного представления Прейсака. В заключение, обсуждается влияние формы размагничивающего поля на относительный вес обратимых и необратимых компонент намагничивания. Получено, что обратимая компонента увеличивается благодаря форме размагничивающего поля. Полученные результаты в этой статье относятся не только к магнитным системам, но также для математического моделирования произвольных гистерезисных систем.

     

Fe-Pb合金凝固多相体系内偏析形成过程的三维数值模拟

基于Eulerian-Eulerian方法和流体体积技术,建立了三维多相流体动力学凝固模型,并将其与质量、动量、溶质和热焓守恒方程相耦合,对Fe-Pb合金侧向凝固过程进行了数值模拟.首先,分析了分布面积二次梯度((SPb))和浓度二次梯度((CPb))对偏析模式的影响,结果表明:液、气两相的流动相变使偏析模式表现为上端X形下端V形,X偏析由气相相变驱动力和多取向相变作用下的"散射"形成;ttc时,随(SPb)和(CPb)曲线降低,X偏析的下偏析角增大,上偏析角和V偏析角减小,Pb收得率增大,有利于获得含量稳定弥散的凝固组织.此外,还研究了液、气两相交互流动下通道偏析的形成机理,结果表明:通道偏析仅存在于流动-相变交互作用(ul·cl和ug·cg)为负值的区域,该区域的流动扰动抑制合金的局部凝固,促进偏析通道生长;流动-相变交互作用负值越小,偏析通道持续增长越稳定.模拟结果与实验结果相符合,验证了模型的准确性.     

Mathematical modelling of magnetically targeted drug delivery

A mathematical model for targeted drug delivery using magnetic particles is developed. This includes a diffusive flux of particles arising from interactions between erythrocytes in the microcirculation. The model is used to track particles in a vessel network. Magnetic field design is discussed and we show that it is impossible to specifically target internal regions using an externally applied field.     

Clinotron的初步理论和数值模拟

在理论分析的基础上,模拟研究了微电真空太赫兹器件Clinotron。求解了色散方程,获得该器件的基本结构参数和性能参数;在考虑空间电荷场和电子回旋运动的情况下,得出能使器件内片状束稳定传输的周期永磁聚焦的磁场条件;利用3维全电磁粒子模拟程序对Clinotron进行了初步模拟以及优化。数值模拟结果表明：当输入电压为4.5 kV、发射束流大小为190 mA、输入电功率为855 W时, 运行频率为218 GHz的Clinotron的输出平均功率40 W,功率转换效率达4.68%。     

Clinotron的初步理论和数值模拟

在理论分析的基础上,模拟研究了微电真空太赫兹器件Clinotron。求解了色散方程,获得该器件的基本结构参数和性能参数;在考虑空间电荷场和电子回旋运动的情况下,得出能使器件内片状束稳定传输的周期永磁聚焦的磁场条件;利用3维全电磁粒子模拟程序对Clinotron进行了初步模拟以及优化。数值模拟结果表明:当输入电压为4.5 kV、发射束流大小为190 mA、输入电功率为855 W时, 运行频率为218 GHz的Clinotron的输出平均功率40 W,功率转换效率达4.68%。     

Theoretical analysis and numerical simulation of development length of straight steel fiber in cementitious materials

In fiber-reinforced concrete, it is important to choose an appropriate length in each fiber to develop its full yield strength without a failure in the bond strength between the fiber and the concrete. This length is called the fiber development length, L_df. The bond capacity is evaluated between the fiber and the concrete using the pull-out tests. This test evaluates the bond capacity of various types of steel fiber surfaces relative to a specific embedded length. If the steel fiber is smooth and straight, the distribution of tensile stresses will be uniform around the fiber at a specific section and varies along the anchorage length of the fiber and at a radial distance from the surface of the fiber. Pull-out tests can be performed on an embedded straight steel fiber in concrete matrix, in this case, the tensile force, P, is increased gradually and the number of cracks and their spacings and widths is recorded. The bond stresses vary along the fiber length between the cracks. The strain in the steel fiber is maximum at the cracked section and decreases toward the middle section between cracks. If the embedded length of the straight steel fiber is greater than the development length, the steel fiber may yield, leaving some length of the fiber in the concrete. The linear elastic behavior of the fiber-matrix system is interrupted by interface debonding which occurs due to overall weak bonding between the concrete matrix and the surface of the steel fiber. This paper introduces new developed shear lag model and explains simplified method to find the development length of straight steel fiber in concrete matrix using finite element model and analysis of shear lag stresses, where the maximum tension force which is applied on the steel fiber is resisted by another internal force related with the ultimate average bond stress, steel fiber diameter and its yield strength.     

衍射图样的数值模拟

介绍了平面波角谱理论中得到衍射积分公式的方法.用Matlab计算了15×15矩孔阵列近场区域的衍射图样,在距离周期衍射屏特定距离处出现衍射屏的自成像.     

Mathematical modelling of thermophysical processes at peat firing and smoldering

A new statement and the numerical solution of the problem of the peat layer firing as a result of the effect of the surface fire is given on the basis of a mathematical model of a porous reacting medium. The original reagent smoldering at moderate temperatures (T ₁ ≤ 750 K) is found to be determined by the processes of heat and mass exchange with the forest fire source, peat drying and pyrolysis, reaction of the carbon oxide oxidation, thermophysical characteristics, and the peat height as well as by the thickness of the water layer under the peat layer.     

三角调频连续波建模与数值仿真

三角调频连续波由于测距精度高,所以在测距系统中有着广泛的应用。结合模数函数、绝对值函数和符号函数,推导了三角调频连续波的频率和相位的数学表达式,并进行了建模及仿真。仿真结果表明,频率和相位的数学表达式满足微积分的关系,能够描述三角波调频在跨越调频周期时的不连续性。     

Design and numerical simulation of novel DBRs

In this paper, a numerical simulation of the traditional graded distributed Bragg reflector (DBR) and a design of the novel DBR with short period superlattices (SPSs DBR) used by vertical cavity surface emitting laser (VCSEL) are reported. First, the optical characteristic matrix of the graded DBRs is derived using the theories of thin film optics. Second, its reflective spectrum is numerical simulated and it is found that the simulative results are similar with the experimental data. The difference of the cavity mode position between the experimental and simulative data is discussed. Finally, based on the simulative results of graded DBR, a novel DBR with 4.5-pair GaAs/AlAs SPSs is designed, and its reflective spectrum is numerical simulated and analyzed.     

Evaluation of deconvolution modelling applied to numerical combustion

A possible modelling approach in the large eddy simulation (LES) of reactive flows is to deconvolve resolved scalars. Indeed, by inverting the LES filter, scalars such as mass fractions are reconstructed. This information can be used to close budget terms of filtered species balance equations, such as the filtered reaction rate. Being ill-posed in the mathematical sense, the problem is very sensitive to any numerical perturbation. The objective of the present study is to assess the ability of this kind of methodology to capture the chemical structure of premixed flames. For that purpose, three deconvolution methods are tested on a one-dimensional filtered laminar premixed flame configuration: the approximate deconvolution method based on Van Cittert iterative deconvolution, a Taylor decomposition-based method, and the regularised deconvolution method based on the minimisation of a quadratic criterion. These methods are then extended to the reconstruction of subgrid scale profiles. Two methodologies are proposed: the first one relies on subgrid scale interpolation of deconvolved profiles and the second uses parametric functions to describe small scales. Conducted tests analyse the ability of the method to capture the chemical filtered flame structure and front propagation speed. Results show that the deconvolution model should include information about small scales in order to regularise the filter inversion. a priori and a posteriori tests showed that the filtered flame propagation speed and structure cannot be captured if the filter size is too large.     

Mathematical simulation of high-power laser systems

The energetic, spectral, and time characteristics of a high pressure CO₂ laser at high pumping level are studied. It is shown that in the free generation regime high power radiation pulses R~ 10^–8 sec in duration with a multifrequency spectrum can be obtained. Laser output characteristics are presented as functions of gas pressure, pumping energy, and resonator parameters.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 15–20, June, 1981.     

Mathematical simulation of the femtosecond-laser inscription of optical waveguides

Based on the hierarchy of mathematical models with regard to parallel algorithms and effective systems, the interaction of a femtosecond-laser pulse with a transparent dielectric medium (fiber-optic glass) is numerically simulated. The features of this interaction are studied for the initial Gaussian and ring distributions of the laser beam. The critical powers at which a local collapse of the beam takes place are determined for the Schrödinger equation. A comparison of the results obtained using various models is presented.     

Mathematical modelling of ultrasonic wave scattering by sub-surface cracks

Scattering of ultrasonic signals by a sub-surface crack in a specimen which is immersed in a water bath has been analysed on the basis of a mathematical model, which accounts for: (1) signal transmission across the water-solid interface; (2) partial illumination of the crack; (3) the relaiton between the crack-opening displacement and the received transducer signal; and (4) an accurate mathematical computation of the crack-opening displacement in the frequency domain. The modelling from the specified initial electrical pulse to the measured scattered signal is completed by converting the frequency-domain results to the time domain. Results are presented for both the pulse-echo and the pitch-catch configuration, and for a number of crack orientations and crack depths.