114,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Addresses the very topical, crucial and original subject of parameter identification and optimization within multiscale modeling methods
Multiscale Modelling and Optimization of Materials and Structures presents an important and challenging area of research that enables the design of new materials and structures with better quality, strength and performance parameters as well as the creation of reliable models that take into account structural, material and topological properties at different scales. The authors’ approach is four-fold; 1) the basic principles of micro and nano scale modeling techniques; 2) the connection of micro and/or nano scale models with macro simulation software; 3) optimization development in the framework of multiscale engineering and the solution of identification problems; 4) the computer science techniques used in this model and advice for scientists interested in developing their own models and software for multiscale analysis and optimization.
The authors present several approaches such as the bridging and homogenization methods, as well as the general formulation of complex optimization and identification problems in multiscale modelling. They apply global optimization algorithms based on robust bioinspired algorithms, proposing parallel and multi-subpopulation approaches in order to speed-up computations, and discuss several numerical examples of multiscale modeling, optimization and identification of composite and functionally graded engineering materials and bone tissues. Multiscale Modelling and Optimization of Materials and Structures is thereby a valuable source of information for young scientists and students looking to develop their own models, write their own computer programs and implement them into simulation systems.
- Describes micro and nano scale models developed by the authors along with case studies of analysis and optimization
- Discusses the problems of computing costs, efficiency of information transfer, effective use of the computer memory and several other aspects of development of multiscale models
- Includes real physical, chemical and experimental studies with modern experimental techniques
- Provides a valuable source of information for young scientists and students looking to develop their own models, write their own computer programs, and implement them into simulation systems.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 557
Veröffentlichungsjahr: 2022
Ähnliche
Multiscale Modelling and Optimisation of Materials and Structures
Tadeusz Burczyński
Institute of Fundamental Technological Research
Polish Academy of Sciences
Warszawa, Poland
Maciej Pietrzyk
AGH University of Science and Technology
Kraków, Poland
Wacław Kuś
Silesian University of Technology
Gliwice, Poland
Łukasz Madej
AGH University of Science and Technology
Kraków, Poland
Adam Mrozek
AGH University of Science and Technology
Kraków, Poland
Łukasz Rauch
AGH University of Science and Technology
Kraków, Poland
This edition first published 2022© 2022 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Tadeusz Burczyński, Maciej Pietrzyk, Wacław Kuś, Łukasz Madej, Adam Mrozek, and Łukasz Rauch to be identified as the authors of this work has been asserted in accordance with law.
Registered OfficesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
Editorial Office111 River Street, Hoboken, NJ 07030, USA
For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.
Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats.
Limit of Liability/Disclaimer of WarrantyThe contents of this work are intended to further general scientific research, understanding, and discussion only and are not intended and should not be relied upon as recommending or promoting scientific method, diagnosis, or treatment by physicians for any particular patient. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of medicines, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each medicine, equipment, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging‐in‐Publication data applied for
Hardback ISBN: 9781119975922
Cover Design: WileyCover Image: © Dr Adam Mrozek
Preface
Multiscale Modelling and Optimisation of Materials and Structures presents an important and challenging area of research that enables the design of new materials and structures with better quality, strength, and performance parameters. It also provides a possibility for the creation of reliable models that take into account structural, material, and topological properties at different scales. The book addresses four major areas: (i) the basic principles of macroscale, microscale, and nanoscale modeling techniques; (ii) the connection of microscale and/or nanoscale models with macrosimulation software; (iii) optimization development in the framework of multiscale engineering and the solution of identification problems; (iv) the computer science techniques used in this class of models and advice for scientists interested in developing their own models and software for multiscale analysis and optimization.
Therefore, the book presents several approaches to multiscale modelling, such as the bridging and homogenization methods, as well as the general formulation of complex optimization and identification problems in multiscale simulations. It also presents the application of global optimization algorithms based on robust bioinspired algorithms, proposes parallel and multi‐subpopulation approaches in order to speed‐up computations, and discusses several numerical examples of multiscale modeling, optimization, and identification of composite and functionally graded engineering materials.
Multiscale Modelling and Optimisation of Materials and Structures is thereby a valuable source of information for young scientists and students looking to develop their own models, write their own computer programs, and implement them into simulation systems.
Biography
Tadeusz Burczyński: His expertise is in computational sciences, including computational intelligence, computational mechanics, and computational materials science, especially in optimization and multiscale engineering.
Maciej Pietrzyk: His research focuses on numerical modelling including multiscale approach and the application of optimization techniques in materials science.
Wacław Kuś: His scientific interests are related to the applications of parallel and HPC methods in the optimization of multiscale problems in mechanics and biomechanics.
Łukasz Madej: His expertise is in computational materials science and process engineering. The main area of interest is full‐field multiscale modelling of industrial processes and phenomena.
Adam Mrozek: His research focuses on molecular dynamic simulations, optimization of mechanical properties of the 2D materials, and multiscale modelling.
Łukasz Rauch: The main interest of his research is focused on computer science applied in industry including conventional way of modelling as well as application of surrogate models.
1Introduction to Multiscale Modelling and Optimization
A wide selection of materials exhibits unusual in‐use properties gained by control of phenomena occurring in mesoscale, microscale, and nanoscale during manufacturing. Examples of such materials range from constructional steels (e.g. AHSS – Advanced High Strength Steels for automotive industry [8] and titanium alloys for aerospace industry [10]) through various materials for energy applications [6] to new biocompatible materials for ventricular assist devices [9] or other biomedical applications [11]. Due to potential advances in materials science that could dramatically affect the most innovative technologies, further development in this field is expected. For this to happen, materials science has to be supported by new tools and methodologies, among which numerical modelling plays a crucial role.
On the other hand, to predict the correlation between processing parameters and product properties properly, one needs to investigate macroscopic material behaviour and phenomena occurring at lower dimensional scales, at grain level or even at atomistic levels. Thus, multiscale modelling with the digital materials representation (DMR) concept [7] is a research field that potentially can support the design of new products with unique in‐use properties. The development of new materials modelling techniques that tackle various length scale phenomena is observed in many leading research institutes and universities worldwide. Multiscale analysis of length and temporal scales has already found a wide range of applications in many areas of science. Advantages provided by a combination of numerical approaches: finite element (FEM), crystal plasticity finite element (CPFEM), extended finite element (XFEM), finite volume (FVM), boundary element (BEM), meshfree, multigrid methods, Monte Carlo (MC), cellular automata (CA), molecular dynamics (MD), molecular statics (MS), phase field and level set methods, fast Fourier transformation, etc. are already being successfully applied in practical applications [14].
This book's main feature, which distinguishes it from other publications, is that it is focused on modelling of processing of materials and that it combines the problem of multiscale modelling with the optimization tasks providing a wide range of possibilities for practical industrial applications.
The first part of the book contains a presentation of basic principles of the microscale and nanoscale modelling techniques. The second part supplies information about applications of optimization and identification techniques in multiscale modelling. The book is recapitulated by presenting information on the computer science techniques used in multiscale modelling, and it is focused on computer implementation issues, advising scientists interested in developing their multiscale models.
1.1 Multiscale Modelling
1.1.1 Basic Information on Multiscale Modelling
During the last years, numerical modelling became a widely used tool that successfully supports and comprehends experimental research on various modern materials. Basic principles of this modelling technique, as well as the classification of models, can be found in fundamental works of Allix [1] or Fish [4], and Horstemeyer and Wang [5] which discussed possible scopes of applications of multiscale modelling in the industry. However, within the last several years, various papers on multiscale modelling have been published in the scientific literature concerning both theoretical issues and practical applications. A variety of problems was raised in these papers, and several ambiguities connected with nomenclature and definitions can be found. The problem of naming various scales in multiscale modelling is an example of a lack of consistency in definitions. Commonly, ‘macro’, ‘meso’, ‘micro’, and ‘nano’ scales are mentioned (Figure 1.1).
Nevertheless, this nomenclature can be misleading and often not sufficient. The term ‘macroscale’ is used for different scales, from large structures to single crystals. Others describe the single‐crystal level as ‘microscale’. In many cases, the characteristic entities in the particular scale are important, not the characteristic length. For example the characteristic length of the scale, in which grains are distinguishable, can vary from nanometres to centimetres. Moreover, the same grain‐level scale can be applied as macroscale in connection with an atomistic scale model or as microscale with a macroscopic model of a large structure deformation. Therefore, this book uses the term ‘coarse’ scale for the scale with greater characteristic length and ‘fine’ scale for the scale with smaller characteristic length. It allows using the terms ‘coarser’ and ‘finer’ to describe relations between more than two scales. To keep the concept clear, the fine or coarse scales are usually related to the governing structures in particular cases (e.g. grains, atoms). Despite the length scale problem, there is also an issue with the time scale as many of the phenomena occur in significantly different time regimes. Usually, the temporal scale is unified across the different length scales to provide physically relevant results, which sometimes makes the computational model expensive.
Figure 1.1 Multiscale concept diagram which illustrates ‘macro’, ‘meso’, ‘micro’, and ‘nano’ scales.
The lack of a definition for the term multiscale in the numerical modelling community is another problem. The broad understanding of multiscale modelling includes all problems with important features in multiple scales (temporal and spatial) [15]. Stricter definitions are also used. In the narrow sense of the term, modelling is multiscale when the models at different scales are simultaneously considered [13]. There is a fundamental difference between these two definitions. Important features at more than one scale are present in almost all problems of material science. A microstructural base of work hardening is one of the most common examples. Hardening is a process at a clearly different scale than plastic deformation modelled with a continuity assumption. Hence we can describe the simulation of plastic deformation and work hardening as the multiscale model. One can also define the multiscale model as a system of submodels at two or more scales but not necessarily simultaneously computed. Some examples are discussed in the book. Two submodels must work simultaneously when both are dependent on results from the second one. Such models are two‐way coupled, contrary to one‐way coupling where one of the models is independent.
The range of possible applications of multiscale modelling is extensive. It includes such different topics as social issues, economy, ecosystems, weather forecast, and physical and chemical processes occurring in gases, fluids, and solids. Material models are an essential branch of multiscale modelling, combining physical and chemical processes, mainly in solids. Such a wide variety of topics makes revising all multiscale modelling applications impossible. Even in material science, several families of applications can be distinguished. The main areas are modelling of polymers, composites, solidification, thin layers, or polycrystalline materials. Each of these areas has its own characteristic features, distinguishing them from the others. While the bases are common for all areas, possible solutions to particular problems are different.
In this book, applications of multiscale modelling to processes occurring in polycrystalline materials, mainly metals and their alloys, are reviewed and addressed. The most distinguishing feature of metals and their alloys is their unrepeatable microstructure. However, the atomistic structure of crystals is regular, while properties of these materials depend mostly on imperfections of atomic structures (dislocations, solid solutions of atoms, grain boundaries, etc.). The mesoscale picture of the material structures includes disordered (usually) distribution of grain shapes, sizes and orientations, as well as strong anisotropy of single crystals and grain boundaries. It is a significant difference in comparison with composites and polymers (with its repeatable microstructure). Then, the difference between the modelling of crystallization and thin layers deposition lies within lack or a very small amount of fluids. It must be remembered that modelling of fluids flow is governed by different types of equations. Moreover, numerical techniques for microscale models and coupling of scales are different in the presence of fluids. Due to this diversity in materials and different techniques used for their description, the book focuses mostly on mentioned metals and their alloys to provide in‐depth information on the practical application of dedicated multiscale modelling and optimization techniques.
1.1.2 Review of Problems Connected with Multiscale Modelling Techniques
As mentioned, in the field of metallic materials, several classifications of multiscale models exist. They depend on such issues as the character of coupling between the scales or methodology of data transformation between the scales. Coupling between the scales can be either strong or weak. Strong coupling combines a description of both (or more) scales into one equations system. In weak coupling, only some data are transferred between the subsequent scales. In the fine scale, the boundary or initial conditions can be developed based on coarse scale data (top‐down approach/downscaling), while data from the fine scale can be used as material properties or a material state in the coarse scale (bottom‐up approach/upscaling). Weakly coupled models have a relatively flexible structure and can be classified into top‐down and bottom‐up approaches.
The strength of coupling has some significant consequences. The strongly coupled models are usually faster and have a better mathematical and theoretical background. Usually, such models are solved with a single numerical method. However, phenomena in all scales must be described with consistent mathematical formulations (usually partial differential equations), which is rather hard or even not possible for some phenomena characterized, e.g.by stochastic behaviour. Moreover, the strongly coupled submodels cannot be separated, and their parts cannot be replaced with other submodels, which make their adaptation difficult. The weakly coupled models are more flexible, from both mathematical and numerical points of view, which makes their development and adaptation much easier. The coupling strength is linked with a methodology of data transfer between the scales.
Various classifications of the multiscale modelling methods were proposed having in mind all the aforementioned features of these techniques. Since this book is dedicated to simulations of materials processing methods, the multiscale models are classified into two groups: upscaling and concurrent approaches. In the upscaling class of methods, constitutive models at higher scales are constructed from observations and models at lower, more elementary scales [2]. By a sophisticated interaction between experimental observations at different scales and numerical solutions of constitutive models at increasingly larger scales, physically based models and their parameters can be derived at the macroscale. In this approach, it is natural that the microscale problem has to be solved at several locations in the macro‐model. For example, in the application of the concept of computational homogenization for each integration point of finite elements in the macroscale, the representative volume element (RVE) is considered in the microscale (Figure 1.2).
Figure 1.2 Illustration of computational homogenization concept.
Figure 1.3(a) Internal variable method as a precursor of upscaling and (b) the idea of the CAFE model.
Thus, classification of the multiscale problems with respect to the upscaling goals and the upscaling costs is needed, which is one of the book's objectives. The idea of upscaling has, in an intuitive manner, been used in engineering for decades. A commonly used approach based on a calculation of the flow stress in the FE model by a solution of the differential equation for the evolution of dislocation populations (Figure 1.3a) can be considered the precursor of upscaling.
Significant progress was made when more advanced discrete methods were used in the microscale; see Figure 1.3b, where the CAFE method (cellular automata finite element) is shown as an example. Now we witness further development of multiscale methods in computational science, which couple fine and coarse scales more systematically. A variety of models and methods is used in both microscale/nanoscale and macroscale. The models are generally different for the two scales, but solutions using the same model in the coarse and fine scales are common. FE2 method is a typical example of such an approach [3].
In concurrent multiscale computing, one strives to solve the problem simultaneously at several scales (in practice two‐scales) by an a priori decomposition of the computational domain. Large‐scale problems are solved, and local data (e.g., displacements) are used as boundary conditions for a more detailed part of the problem. The question of how the fine scale is coupled to the coarse scale is essential in this approach. The major difficulty in coupling occurs when different models describe fine scales and coarse scales, for example coupling FE to MD. In other words, the objective is to find a computationally inexpensive but still accurate approach to the decomposition problem. The terms ‘homogenization’ and ‘localization’ are commonly used for operations during data transfer. Again, this terminology is not well defined. Homogenization, in multiscale modelling, has two meanings. In the context of strongly coupled models, based on differential equations, it is understood as the asymptotic homogenization, the method of studying partial differential equations with rapidly oscillating coefficients. The in‐depth description of the homogenization and the averaging based on the perturbation theory, as well as their application to multiscale modelling, can be found e.g.in [12]. However, most authors use this term differently, describing it as ‘any method of transforming a heterogeneous field into a homogeneous one’. Weighted averaging is usually used; however, more advanced techniques are also present. In this book, we use the term homogenization in common understanding and asymptotic homogenization in the sense of the element of perturbation theory. Details on the multiscale classification and examples of its applications can be found in Chapter 5 of the book.
1.1.3 Prospective Applications of the Multiscale Modelling
In recent years, a gradual paradigm shift has been taking place in the selection of materials to suit particular engineering requirements, especially in high‐performance applications. The empirical approach adopted by materials scientists and engineers in choosing materials parameters from a database is being replaced by design based on the DMR concept. Features that span across a large spectrum of length scales are altered and controlled to achieve the desired properties and performance at the macroscale. Research efforts, in this aspect, include the development of engineering materials by changing the composition, morphology, and topology of their constituents at the microscopic/mesoscopic level. The objective of this book is to show multiscale methods and their applications in computational materials design. From one side, we present computational multiscale material modelling based on the bottom‐up/top‐down, one‐way coupled description of the material structure in different representative scales. On the other side, our intention is to show possibilities of a combination of multiscale methods with optimization techniques.
1.2 Optimization
Solution of optimization problems in multiscale modelling allows finding structures with better performance or strength in one scale with respect to design variables in another scale. In this case, the typical situation is to find a vector of material or geometrical parameters on the micro‐level, which minimizes an objective function dependent on state fields on a macro‐level of the structure.
A special case of optimization problems associated with the multiscale approach is the optimization of atomic clusters for the minimization of the system's potential energy. This case has an important consequence, especially in the design of new 2D nanomaterials and nanostructures.
The identification problem is formulated as the evaluation of some geometrical or material parameters of structures in one scale having measured information in another scale. The important case of identification in multiscale modelling is to find material properties, the shape of the inclusions/fibres or voids in the microstructure having measurements of state fields made on the macro‐object. The identification problem is formulated and considered as a special optimization task.
The analysis methods of multiscale models based on computational homogenization are adopted for these classes of problems. To solve optimization and identification problems, global optimization methods based on bioinspired algorithms are used.
1.3 Contents of the Book
The aspects of multiscale modelling mentioned previously have already been discussed in numerous publications, including several books [4, 12, 13, 15]. However, multiscale modelling still remains a difficult task, and its valid and reliable application is quite difficult. Moreover, the lack of an unambiguous definition leads to misunderstandings and mistakes. The book supplies some practical information concerning the development and application of multiscale material models, in particular in combination with optimization techniques.
As mentioned, the book is divided into three main sections. The first section is composed of Chapters 2 and 3, and it is focused on discussion of phenomena occurring in materials in processing and on models used to describe these phenomena. Such phenomena as recrystallization, phase transformations, cracking, fatigue, and creep are discussed very briefly. The modelling methods are divided into two groups. The first includes computational methods for continuum such as FEM, XFEM, and BEM. Discrete methods describing microscale and nanoscale phenomena include MS as well as MD, CA, and MC approaches. Computational homogenization methods, which are used in the coupled multiscale models, are also discussed in this part of the book. Basic principles of methods of optimization are also described in this part of the book, with a particular emphasis on the methods inspired by nature.
The second section of the book is composed of Chapters 4 and 5 and focuses on DMR. Case studies based on DMR are presented. Applications of methods described in Chapters 2 and 3 to modelling various processes and phenomena are shown.
The third section of the book is connected with applications of multiscale optimization methods. Such issues as optimization of atomic clusters, material parameters optimization, shape optimization, and topological optimization are discussed. The problem of identification in multiscale modelling is also presented.
Computer implementation issues for multiscale models are recapitulated in the book. Implementation of selected algorithms concerning visualization as well as scales coupling with use of commercial software are described in Chapter 7. The part will highlight the possibilities of increasing the computational efficiency, which is especially important when optimization based on a direct problem model of multiscale nature is considered.
References
1
Allix, O. (2006). Multiscale strategy for solving industrial problems. In:
III European Conference on Computational Mechanics
(ed. C.A. Motasoares, J.A.C. Martins, H.C. Rodrigues, et al.), 107–126. Dordrecht: Springer.
2
De Borst, R. (2008). Challenges in computational materials science, multiple scales, multi‐physics and evolving discontinuities.
Computational Material Science
43: 1–15.
3
Feyel, F. (1999). Multiscale FE2 elastoviscoplastic analysis of composite structures.
Computational Materials Science
16: 344–354.
4
Fish, J. (2014).
Practical Multiscaling
. Wiley.
5
Horstemeyer, M.F. and Wang, P. (2003). Cradle‐to‐grave simulation‐based design incorporating multiscale microstructure‐property modeling: reinvigorating design with science.
Journal of Computer‐Aided Materials Design
10: 13–34.
6
Kilner, J.A., Skinner, S.J., Irvine, S.J.C., and Edwards, P.P. (ed.) (2012).
Functional Materials for Sustainable Energy Applications
. Oxford, Cambridge, Philadelphia, New Delhi: Woodhead Publishing Ltd.
7
Madej, L., Wang, J., Perzynski, K., and Hodgson, P.D. (2014). Numerical modeling of dual phase microstructure behavior under deformation conditions on the basis of digital material representation.
Computational Materials Science
95: 651–662.
8
Matlock, D.K., Krauss, G., and Speer, J.G. (2005). New microalloyed steel applications for the automotive sector.
Materials Science Forum
500–501: 87–96.
9
Milenin, A. and Kopernik, M. (2011). Microscale analysis of strain‐stress state for TiN nanocoating of POLVAD and POLVAD_EXT.
Acta of Bioengineering and Biomechanics
13: 11–19.
10
Muszka, K., Lopez‐Pedrosa, M., Raszka, K. et al. (2014). The impact of strain reversal on microstructure evolution and orientation relationships in Ti‐6Al‐4V with an initial alpha colony microstructure.
Metallurgical and Materials Transactions A
45: 5997–6007.
11
Niinomi, M. (2002). Recent metallic materials for biomedical applications.
Metallurgical and Materials Transactions A
33: 477–486.
12
Pavliotis, G. and Stuart, A. (2008).
Multiscale Methods: Averaging and Homogenisation
. Springer.
13
Weinan, E. (2011).
Principles of Multiscale Modeling
. Cambridge University Press.
14
Yanagimoto, Y., Banabic, D., Banu, M., and Madej, L. (2022).
Simulation of Metal Forming – Visualization of Invisible Phenomena in the Digital Era
. CIRP Annals Manufacturing Technology in press.
15
Yip, S. (ed.) (2005).
Handbook of Materials Modeling
. Dordrecht: Springer.
2Modelling of Phenomena
The major physical phenomena responsible for material behaviour under manufacturing and exploitations stages are described in this chapter. It highlights the limitations of conventional modelling approaches and introduces possibilities provided by the multiscale solutions. Both nanoscale and microscale phenomena, including quantum scale, are discussed, and nanostructures and microstructures are analyzed.
2.1 Physical Phenomena in Nanoscale
The laws of quantum mechanics underlie the basis of the description of physical phenomena, as well as simulation methods in the nanoscale. However, the solution of the Schrödinger equation – the fundamental equation which describes the behaviour of the particles in quantum scale – even the numerical one, is restricted to the relatively simple cases of molecular systems. The high complexity of the Schrödinger equation and the high dimension of the space in which it is defined exclude the pure ab initio methods [70] from the mechanical engineering computations presented in this book. Therefore, a series of subsequent simplifications and approximations procedures are used to overcome this problem and make the molecular simulations possible on the desired level of accuracy. The quantum mechanics computations, such as methods based on the density functional theory (DFT) [23, 94], are also beyond the scope of this book.
This chapter describes the route of the derivation of classical molecular dynamics (MD) [33, 38, 103], based on the Newtonian equations of motion, from the Schrödinger wave equation. Presented methodology and hierarchy of introduced simplifications also reveal the basis of the formulation of atomic potentials – the simplified models of the interatomic interactions. The problems with a proper approximation of the interactions, which determine behaviour of the atoms in classical MD, underlie the development of the mixed, quantum‐classical approaches (like Car‐Parinello MD [12, 93]), where the motion of the nuclei is determined by the Newtonian equation of motion, while the elements of the electronic structure are still treated as quantum objects. Such approaches allow estimation of the potential energy and interatomic forces on the fly in each MD integration step without engaging any predefined potentials. However, in the case of large mechanical computations at the nanoscale (up to billions of atoms), the classical MD method still plays an important role [101]; thus, the most popular atomic potential models are presented in the second section of this chapter.
2.1.1 The Linkage Between Quantum and Classical Molecular Mechanics
In quantum mechanics, behaviour of the particles (nuclei, electrons, etc.) is described by the Schrödinger equation. Unlike the classical equation of motion, the solution of the Schrödinger equation provides probabilistic information about the quantum mechanical system.
The time‐dependent wave function Ψ of a quantum system with i electrons and I nuclei has a general form:
where: vectors ri and Ri denote, respectively, the electronic and nuclear coordinates in the three‐dimensional space. In this chapter, the abbreviated notation (r and R) is used. The probability of finding the quantum system in the elementary volume element dV of the considered configuration space centred at the point (r, R) at time t is expressed as:
The full wave function (2.2) can be obtained as the solution of the time‐dependent Schrödinger equation:
The standard Hamiltonian H of the quantum system is given as a sum over potential and kinetic energy operators:
where: Te and TN denote kinetic energies of the electrons and the nuclei, respectively:
The operators: Vee, VeN,, and VNN are the potential energies of the interactions (Coulomb potentials) between the electrons, the nuclei, and between the electrons and the nuclei, respectively:
where: ZI and MI refer to the atomic number and mass of the I‐th nucleus, and me and e denote mass and charge of the electron, respectively.
One of the possible approaches of the derivation of classical MD, according to [33, 71, 128], utilizes the equations of the time‐dependent self‐consistent field (TDSCF) method. These equations can be obtained by decomposition of the Hamiltonian (2.4) and subsequent separation of the nuclear and electronic contributions to the wave function (2.1).
The full Hamiltonian operator (2.4) can be rearranged to a simpler form:
where electronic Hamiltonian He is defined as follows:
and may be written as:
The term Ve corresponds to the total potential energy of the considered system:
The full wave function, given in the form (2.1), depends on the electronic as well as nuclear coordinates (also known as fast and slow coordinates, respectively). These two contributions have to be separated. The traditional way is the approximation of the full wave function with the following product form:
This kind of simplification refers to the one‐determinant or single‐configuration version of the full wave function. The electronic ϕ(r,t) and nuclear χ(R,t) wave functions should be normalized to unity for any time point t, so the following conditions should be satisfied:
The term in the formulation (2.14) is the phase factor, specified as:
The Schrödinger equation for the electronic wave function ϕ(r,t) might be obtained by substituting approximation (2.14) into the Schrödinger Eq. (2.3) with the full Hamilton operator (2.4). After the multiplication of (2.3) from the left by 〈φ|, 〈χ|, and assuming energy conservation:
the following coupled system of equations can be formulated:
Equations (2.18) and (2.19) are the fundamentals of the TDSCF method, developed by Dirac [21]. This is a mean‐field method, i.e. the nuclei (‘slow’ degrees of freedom) move in the average field of the electrons (the ‘fast’ ones) and vice versa. Both types of degrees of freedom are coupled together, and the proper feedback in the two directions is assured. It can be noted that mean‐field description is a price to pay for the simplest separation of electronic and nuclear coordinates [71].
Since the TDSCF method is still based on quantum‐mechanical equations, other approximations and simplifications must be introduced to reveal the linkage between quantum and classical MD. In the subsequent step, the nuclei should be treated as classical point particles. It can be done by rewriting the nuclear wave function as:
The amplitude and phase factors, denoted by A and S, respectively, should be both real and positive in the polar representation. In the next stage, the nuclear wave function (2.20) is substituted into Eq. (2.19), and after splitting the real and imaginary parts, the coupled equations for the nuclei expressed in the variables A and S can be obtained:
Equations (2.21) and (2.22) formulate quantum fluid dynamics representation [71] and can be used to solve the time‐dependent Schrödinger equation. In the further process of derivation of the classical MD, Equation (2.21) plays an even more important role. The right side of this equation directly depends on ħ, but this term can be neglected when the classical limit (ħ →0) is applied:
After neglecting the terms proportional to ħ, the newly obtained equation is isomorphic to the Hamilton‐Jacobi formulation of equations of motion – a well‐known form of the classical mechanics:
The Hamiltonian operator in Eq. (2.24) is now defined in terms of generalized coordinates R=(R1…RI) and their conjugated momentum P=(P1…PI):
where: T(P) and V(R) are appropriate kinetic and potential energy operators, respectively. The momentum PI of the I‐th nuclei and associated force is defined as follows:
Now, according to formulation (2.23), the Newtonian equations of motion can be formulated in the following way:
or
At this stage, the motion of the nuclei is described by the familiar law of classical Newtonian mechanics (2.29) in the field of the effective potential (the Ehrenfest potential) generated by the electronic system. This kind of potential is a function of only ‘slow’ (nuclear) coordinates R at time point t and can be interpreted as a result of averaging He over the ‘fast’ (electronic) degrees of freedom, i.e. computing the quantum expectation value of the Hamiltonian He at fixed, current positions of the nuclei.
It can be noted that the electronic Eq. (2.18) of the TDSCF method is still expressed in terms of the nuclear wave function χ(R,t). The wave function has to be substituted by the positions of the nuclei. One of the convenient ways to do this is to replace the probability density of the nuclei |χ(R, t)|2 with a product of delta functions ∏Iδ(RI − RI(t)) and apply again the classical limit (ħ →0) to (2.18). The delta functions should be centred at the current positions R(t) of the nuclei, obtained by solving Eq. (2.29). This approach leads to the time‐dependent electronic Schrödinger equation:
Equations (2.29) and (2.30) are the base of the Ehrenfest MD – one of the hybrid quantum‐classical approaches, like Born‐Oppenheimer or Car‐Parinello MD [12, 93, 128]. The nuclei are treated now as slow classical particles which move according to the Newtonian equation of motion (2.29), while the fast electrons are still quantum objects. The Hamiltonian operator He and electronic wave function ϕ now depend on the classical nuclear positions R(t), so the feedback between classical and quantum parts is guaranteed in both directions. Keeping in mind conventional MD, the last problem which has to be solved is the proper approximation of the effective potential function in the classical equation of motion (2.28).
As mentioned earlier, Ehrenfest MD is a mean‐field method derived from the TDSCF approach. However, transitions between electronic states are still possible. The electronic wave function should be written as a linear combination of the base functions ϕj of the electronic states and complex coefficients cj(t) for given time point t:
where one assumes: . The convenient choice for the base functions ϕj
