Stochastic modelling of combat with reinforcement

The homogeneous combat when reinforcements are made at prespecified force levels has been modelled as a continuous-time discrete-state space Markov process. The effect of reinforcement on various combat characteristics, viz., the mean combat duration, the winning probabilities, the expected survivors, etc., is studied using the uniformization method. Two new concepts, i.e., Reinforcement Effectiveness Index and Reinforcement Parity Curve, have also been defined and are illustrated numerically.     

Stochastic birth-and-growth processes modelling crystallization of polymers with spatially heterogeneous parameters

The morphology of a crystalline polymeric material is characterized by the crystallization process, which occurs during the solidification from the liquid state. This process includes a stochastic birth process (nucleation) and the growth of crystals. From a mathematical point of view, the nucleation process is a marked point process with a stochastic intensity depending on the history of the proceses of birth and growth. Here we consider an extension of the classical Avrami–Kolmogorov model to the case of space-time heterogeneous kinetic parameters. Concepts from stochastic geometry are used, which include the so called causal cone. The Markov property of the process is also analysed.     

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.     

Hand-to-hand combat with thousand-digit integrals

In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multi-step uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K′, E and E′. We find that in many cases such integrals can be “experimentally” evaluated in closed form or that intriguing linear relations exist within a class of similar integrals. Discovering these identities and relations often requires the evaluation of integrals to extreme precision, combined with large-scale runs of the “PSLQ” integer relation algorithm. This paper presents details of the techniques used in these calculations and mentions some of the many difficulties that can arise.     

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.     

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.     

Applications of Artificial Neural Networks to combat simulations

As a result of reduced budgets and personnel levels, the Department of Defense has increased reliance on combat simulations for such diverse areas as training, testing, planning, and analysis. Each area has its own set of needs, goals, and objectives for designing future generations of combat simulation models. However, budget constraints alone mandate the development of multipleuse combat models. The bottom line is that future generations of combat models need to be faster, have higher fidelity and larger scale than current models. Research into emerging technologies for approaches to make computer simulations more effective and efficient is an essential ingredient to developing successful future generations of combat models. One emerging technology that has such potential is Artificial Neural Networks (ANN). Potential applications of ANN to combat simulation modeling are discussed. The main results of the author's dissertation Artificial Neural Network Metamodels of Stochastic Computer Simulations [1] are discussed along with the ramifications on combat modeling and recommendations for future research.     

Homomorphisms of heterogeneous algebras

A construction of all homomorphisms of a heterogeneous algebra into an algebra of the same type is presented. A relational structure is assigned to any heterogeneous algebra, and homomorphisms between these relational structures make it possible to construct homomorphisms between heterogeneous algebras. Homomorphisms of relational structures can be constructed using homomorphisms of algebras that are described in [11].     

Analysis of a combat problem: The turret game

The turret game is defined and solved to illustrate the nature of games of combat. This game represents a highly simplified version of air combat, yet it is sufficiently complex so as to exhibit a rich variety of combat phenomena. A review of the formulation of -combat games is included in this paper.on sabbatical leave from Technion, Israel Institute of Technology, Haifa, Israel.     

A verification study of the stochastic salvo combat model

When the stochastic version of the salvo combat model was designed, several assumptions and approximations were made to keep its mathematical structure relatively simple. This paper examines the impact of those simplifications by comparing the outputs of the stochastic model to those from a Monte Carlo simulation across 486 scenarios. The model generally performed very well, even where the battle size was relatively small or the damage inflicted by each missile was not normally distributed. The model’s accuracy did decrease where missiles were positively correlated instead of independent.     

Optimal deployment of logistic units in dynamic combat conditions

One of the problems facing the commander of support units during a combat situation is how to plan the deployment of the units under his command. This paper deals with the tactical problem of how to deploy the support units in order to achieve maximum effectiveness, given the decision of which fighting units will receive support. The problem is formulated as a dynamic decision problem that has some of the characteristics of a dynamic warehouse location problem. A branch and bound technique to reduce the state space of the d.p. formulation is applied in the solution of this model.     

Pensions modelling

Valuable insights into the problem of how to fund defined benefitpension schemes can be obtained by analysis using the standardBlack–Scholes/Merton option pricing model, consideringthe pension fund finances jointly with those of the sponsoringcompany. The nature of the fund assets and liabilities is completelydifferent, and this lies behind current controversies aboutthe appropriate discount rate, valuation, financial accountingand preferential status for pension fund claimants in insolvency.     

The salvo combat model with a sequential exchange of fire

This paper develops a version of the stochastic salvo combat model in which the exchange of fire is sequential, rather than simultaneous. This sequential-fire version is built by modifying the equations in the original simultaneous-fire version. The performance of the sequential model is tested by comparing its outputs with those of a Monte Carlo simulation. The fit between the model and the simulation is very close, especially for the mean and standard deviation of losses. The model is then applied to the Battle of the Coral Sea. The results suggest that attacking first would have given the American force a larger advantage than that provided by an extra aircraft carrier.     

Stochastic upscaling of heterogeneous materials

We propose a probabilistic strategy to upscale the material spatial variability from fine to coarse scale. To implement this idea in a numerical framework, we consider the coarse-scale as stochastic, i. e. its material properties are considered uncertain and modeled as random variables/fields. Numerical examples are shown to demonstrate the applicability of proposed approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hierarchical modeling of heterogeneous plates

The use of asymptotic limits to model heterogeneous plates can be troublesome, because it requires a priori knowledge on the ratio between characteristic lengths of heterogeneities and thickness. Moreover, it also relies on some assumption on the inclusions, like periodicity. We propose and analyze here hierarchical modeling techniques and show that such approach not only avoids such pitfalls, but it is actually simpler to obtain, and it provably converges to the correct asymptotic limits. Its derivation does not requires any restrictive assumptions on the heterogeneities. Copyright © 2014 John Wiley & Sons, Ltd.     

Semiqualitative modelling of bioprocesses

This paper deals with semiqualitative modelling of bioprocesses with a view to their supervision. An analysis of several approaches for modelling shows the difficulties involved in taking into account in a same framework, quantitative and qualitative knowledge, generally available about a process that we want to control. We propose an original approach, placed in the context of semiqualitative modelling, that is supported by a knowledge model the variables and parameters of which are defined by intervals. For these semiqualitative models, we study their properties in simulation and prediction, and more precisely, their fitting based on experimental data. We show that pertinent predictions in a short time can be obtained, making of these semiqualitative models interesting tools for the development of systems for bioprocess supervision     

Mathematical modelling of chip thickness in micro-end- milling: A Fourier modelling

This study developed a model of undeformed chip thickness in micro-end-milling for the use in estimating cutting constants based on measured cutting forces. The proposed estimation method is based upon the invertibility of the average milling force model. In this paper, chip thickness in micro-end-milling was estimated by summing the thicknesses of the conventional chip component and the additional chip component. Thickness was then expressed in terms of Fourier series. The analyses showed that the fast convergence of Fourier series gives the Fourier chip thickness model sufficient accuracy when using only five terms of the truncated Fourier series for common micro-end-milling processes. The Fourier coefficients can be expressed in terms of the ratio of feed per tooth to cutter radius for different numbers of cutter teeth. The accuracy and conciseness of the chip thickness model enables the modelling of average cutting force in a closed form, which can be applied to identify the cutting constants. Cutting force experiments verify that the model prediction agrees very well with the experimental results.