Probabilistic analysis of combat models

This paper reviews recent developments in the field of stochastic combat models. A simple heterogeneous model with attrition rates dependent on the number of surviving forces is considered as a Markov process. Various characteristics of system dynamics are evaluated and expressed in explicit form. Numerical results to illustrate the difference between deterministic and stochastic models are presented. Some areas for further work are pointed out.     

Lanchester model for three-way combat

Lanchester (1960) modeled combat situations between two opponents, where mutual attrition occurs continuously in time, by a pair of simple ordinary (linear) differential equations. The aim of the present paper is to extend the model to a conflict consisting of three parties. In particular, Lanchester’s main result, i.e. his Square Law, is adapted to a triple fight. However, here a central factor – besides the initial strengths of the forces – determining the long run outcome is the allocation of each opponent’s efforts between the other two parties. Depending on initial strengths, (the) solution paths are calculated and visualized in appropriate phase portraits. We are able identify regions in the state space where, independent of the force allocation of the opponents, always the same combatant wins, regions, where a combatant can win if its force allocation is wisely chosen, and regions where a combatant cannot win itself but determine the winner by its forces allocation. As such, the present model can be seen as a forerunner of a dynamic game between three opponents.     

Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries

ABSTRACT

A simple and scalable finite-dimensional model based on the port-Hamiltonian framework is proposed to describe the fluid–structure interaction in tubes with time-varying geometries. For this purpose, the moving tube wall is described by a set of mass-spring-damper systems while the fluid is considered as a one-dimensional incompressible flow described by the average momentum dynamics in a set of incompressible flow sections. To couple these flow sections small compressible volumes are defined to describe the pressure between two adjacent fluid sections. The fluid-structure coupling is done through a power-preserving interconnection between velocities and forces. The resultant model includes external inputs for the fluid and inputs for external forces over the mechanical part that can be used for control or interconnection purposes. Numerical examples show the accordance of this simplified model with finite-element models reported in the literature.     

An analytical model for close combat dynamics

A coherent dynamic combat model is developed from basic principles. The governing set of equations has a striking resemblance to the continuity equation in fluid dynamics with an additional term for the losses of combat units. The salient features of the model are a moving battle front, the replenishment of losses, and the withdrawal of combat units while others are still engaged. A basic example shows that the often used force ratio of three can produce a frontline movement up to 90% of the speed of the attacker. Another example simulates a well documented battle from the American Civil War. It is shown that terrain influences and the absence of reconnaissance had a large adverse effect on the outcome of the battle for the Confederate forces.     

Spatial Lanchester models

Lanchester equations have been widely used to model combat for many years, nevertheless, one of their most important limitations has been their failure to model the spatial dimension of the problems. Despite the fact that some efforts have been made in order to overcome this drawback, mainly through the use of Reaction–Diffusion equations, there is not yet a consistently clear theoretical framework linking Lanchester equations with these physical systems, apart from similarity. In this paper, a spatial modeling of Lanchester equations is conceptualized on the basis of explicit movement dynamics and balance of forces, ensuring stability and theoretical consistency with the original model. This formulation allows a better understanding and interpretation of the problem, thus improving the current treatment, modeling and comprehension of warfare applications. Finally, as a numerical illustration, a new spatial model of responsive movement is developed, confirming that location influences the results of modeling attrition conflict between two opposite forces.     

Semi-dynamic modelling of heterogeneous land combat

We propose a semi-dynamic approach to tactical level land combat modelling from the attacker's viewpoint. Our approach decomposes a battle between heterogeneous forces into stages and mini battles. For each mini battle in a stage, we use three models: a mathematical programming model for optimizing force allocations, a Lanchester simulation model for predicting whether or not the stage targets are reached under the allocations, and a model for weapon effectiveness update from one stage to the next. These models interact with each other within the framework of a decision support system to help the user with allocation decisions as well as prediction of force requirements to win the battle.     

On the influence of design parameters and nonlinearities in vehicle suspensions on road deformation

Among others, two main objectives of modern vehicle design are road friendliness and ride comfort. Both aspects are strongly related since the dynamical tire forces depend on the vertical acceleration of the vehicle. In order to investigate the influence of design and operation parameters, different car models are considered which move with constant velocity on a rippled road. First, a linear half car model is examined and the influence of different design parameters is discussed. Second, nonlinear suspensions with Coulomb friction due to sealings as well as with bilinear shock absorbers are taken into account. The vertical dynamics of the vehicle model and the dynamic tire forces between vehicle and road are calculated using analytical methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于联合作战兵力配置的非合作博弈研究

基于联合作战实体定量分析的作战能力指数组合期望效用,研究了联合作战任务兵力配置问题,建立了二人零和参数博弈模型,并讨论了模型构建的理论依据与算法.     

Firepower concentration in cellular automaton combat models—an alternative to Lanchester

One of a new generation of combat models is examined to determine how its behaviour differs from older approaches based on first-order linear differential equations. This new methodology, which uses ‘cellular automaton’ or ‘agent-based’ models, has been around for a decade, prompting closer scrutiny. The method gives entities within a combat simulation the autonomy to react to circumstances in their local area. The reaction is determined by each entity's ‘personality’. It is found that the automata tend to either fight as a massed force, or form dispersed patterns of clusters within clusters. Such a pattern is known as a ‘fractal’. By adopting this pattern, a non-intuitive relationship between the kill probability of the automata and the force attrition rate develops. This provides a compelling example of how the result presented by earlier workers—that automaton models may evolve into fractal distributions—can have significance for operational researchers.     

Modelling and analysis of dynamics of viral infection of cells and of interferon resistance

Interferons are active biomolecules, which help fight viral infections by spreading from infected to uninfected cells and activate effector molecules, which confer resistance from the virus on cells. We propose a new model of dynamics of viral infection, including endocytosis, cell death, production of interferon and development of resistance. The novel element is a specific biologically justified mechanism of interferon action, which results in dynamics different from other infection models. The model reflects conditions prevailing in liquid cultures (ideal mixing), and the absence of cells or virus influx from outside. The basic model is a nonlinear system of five ordinary differential equations. For this variant, it is possible to characterise global behaviour, using a conservation law. Analytic results are supplemented by computational studies. The second variant of the model includes age-of-infection structure of infected cells, which is described by a transport-type partial differential equation for infected cells. The conclusions are: (i) If virus mortality is included, the virus becomes eventually extinct and subpopulations of uninfected and resistant cells are established. (ii) If virus mortality is not included, the dynamics may lead to extinction of uninfected cells. (iii) Switching off the interferon defense results in a decrease of the sum total of uninfected and resistant cells. (iv) Infection-age structure of infected cells may result in stabilisation or destabilisation of the system, depending on detailed assumptions. Our work seems to constitute the first comprehensive mathematical analysis of the cell-virus-interferon system based on biologically plausible hypotheses.     

An attrition game on a network ruled by Lanchester’s square law

We consider two-person zero-sum attrition games in which an attacker and a defender are in combat with each other on a network. The attacker marches from a starting node to a destination node, hoping that the initial members survive the march. The defender deploys his forces on arcs in order to intercept the attacker. If the attacker encounters the defender on an arc, the attacker incurs casualties according to Lanchester’s square law. We consider two models: a one-shot game in which the two players have no information about their opponents, and a two-stage game in which both players have some information about their opponents. For both games, the payoff is defined as the number of survivors for the attacker. The attacker’s strategy is to choose a path, and the defender’s is to deploy the defending forces on arcs. We propose a numerical algorithm, in which nonlinear programming is embedded, to derive the equilibrium of the game.     

The Role of Metanetworks in Network Evolution

The question of what structures of relations between actors emerge in the evolution of social networks is of fundamental sociological interest. The present research proposes that processes of network evolution can be usefully conceptualized in terms of a network of networks, or “metanetwork,” wherein networks that are one link manipulation away from one another are connected. Moreover, the geography of metanetworks has real effects on the course of network evolution. Specifically, both equilibrium and non-equilibrium networks located in more desirable regions of the metanetwork are found to be more probable. These effects of metanetwork geography are illustrated by two dynamic network models: one in which actors pursue access to unique information through “structural holes,” and the other in which actors pursue access to valid information by minimizing path length. Finally, I discuss future directions for modeling network dynamics in terms of metanetworks.     

一种登陆作战兵力上船装载方案优化计算方法

建立登陆作战兵力上船装载方案优化模型,针对大规模登陆作战兵力上船装载方案确定问题特点,提出一种用需装载兵力、可用舰船和可用泊位类型排列表示方案的方案编码方法,称为方案的类型排列基因码.给出排列基因码生成及进化方法,实现装载方案的遗传优化.计算结果表明,使用该方法可以在较短时间内得到较好的上船装载方案.     

基于仿真的空中进攻战斗效能评估方法研究

通过分析空中进攻战斗的概念定义、兵力编成和过程阶段,构建空中进攻战斗仿真模型,从多次仿真结果中选取战斗效能指标,对效能指标进行简化和标准化处理后,建立空中进攻战斗效能指标体系,在此基础上构建效能指标矢量空间模型和效能优势参数模型,并根据模型对效能指标进行优化分析,最后结合实际算例得出结论.     

Development of a novel family of military campaign simulation models

Dstl has sponsored the development and use of a family of campaign level models of military operations. The models are required to provide an insight into force structure assessment, procurement decisions and doctrinal developments. The family comprises CLARION (land/air war-fighting), COMAND (maritime/air war-fighting) and DIAMOND (non-war-fighting). The two key features of the family are the core role played by the representation of C3I (command, control, communications and information) and the classification of combat units by their interactions with one another at an aggregated level rather than by treating them as a collection of specific combat platforms.     

Modeling and boundary stabilization of a multiple beam system

In this paper we consider two mathematical models for a multiple beam system (MBS) which is composed of two rigidly and angularly connected Euler-Bernoulli beams The cantilevered structure is clamped at one end, and has point controls for forces and bending moments imposed at the other end and at the connection between the two beams The first model incorporates not only transverse deformations of both beams, but also axial compression/extension of the beams. The second model involves only transverse deformations of the beam. By imposing point controls, an unbounded input operator is obtained A variational formulation of the models is used to show well-posedness. Uniform exponential stabilizability of the second model through boundary feedback is established via energy arguments     

Simulation and experimental validation of the dynamic behavior of an electromechanically actuated multiple‐disk clutch

Automated multiple‐disk clutches in automotive drivelines are at present hydraulically or electrohydraulically actuated. The electromechanical clutch actuation represents an alternative. The subject of the investigation is a functional model of an electromechanical clutch actuator which consists of a servo motor, a reduction gear and a mechanism for the realization of the force acting on the disks. For the simulation of the dynamic behavior of the electromechanically actuated multiple‐disk clutch a detailed multibody model is developed which includes models of the electromechanical actuator and the wet multiple‐disk clutch. The multiple‐disk clutch is described by coupled partial models for the rotational and translational dynamics. The coupling parameters are the relative disk distances and the forces between the disks. The results of test bench measurements are compared to simulation results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linearity of multibody dynamic equations with respect to Lagrange multipliers: application to railway dynamics

In this paper, the dynamics of multibody systems with closed kinematical chains of bodies is considered. The main focus is set on non-linearity of the multibody equations with respect to the Lagrange multipliers. When closed chains are considered, loop cutting procedure is a solution to express the constraint equations associated with the loops. Dynamic equations of the multibody tree-like structure are thus completed with the constraint forces via the Lagrange multipliers. In the considered case of railway vehicles, constraints arise from the contact between the rigid wheels and the rails. Corresponding contact forces applied to the wheels appears via the Lagrange multipliers λ and the tangent creep forces as well. Resulting differential-algebraic equations can be transformed into an ODE system and then time-integrated using the coordinate partitioning method [3], when the system is linear with respect to λ. This paper presents an algorithm allowing us to solve this system in case of nonlinearities with respect to λ, which is typical of wheel/rail contact force models. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combat games

We propose a mathematical formulation of a combat game between two opponents with offensive capabilities and offensive objectives. Resolution of the combat involves solving two differential games with state constraints. Depending on the game dynamics and parameters, the combat can terminate in one of four ways: (i) the first player wins, (ii) the second player wins, (iii) a draw (neither wins), or (iv) joint capture. In the first two cases, the optimal strategies of the two players are determined from suitable zero-sum games, whereas in the latter two the relevant games are nonzero-sum. Further, to avoid certain technical difficulties, the concept of a -combat game is introduced.Dedicated to G. LeitmannThe first author wishes to acknowledge the friendship and guidance of George Leitmann, beginning in the author's student days at Berkeley and continuing to the present time. All the authors thank George Leitmann for many recent fruitful discussions on differential games.on sabbatical leave from Technion, Israel Institute of Technology, Haifa, Israel.     

Measuring naval presence using blotto games

Although the combat capabilities of alternative aircraft carrier designs have been much analyzed, force structure comparisons for peacetime uses, such as the projection of presence, are less often quantified. This paper uses a Blotto game framework to evaluate equal cost forces of alternative carrier designs in the presence role.