# A Review of Computational Methods in Materials Science: Examples from Shock-Wave and Polymer Physics

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}) after planar impact load. Inorganic crystalline materials have structural features such as grain boundaries between crystals which are mm to μm in size (cf. Figure 1a), dislocations, and point defects such as vacancies on the atomic scale. Hence, these structures have to be studied from a hierarchical perspective [17].

## 2. Physical and Numerical Modeling

^{−10}m) and typically treat atoms as point particles or spheres with eigenvolume. In principle, the relativistic time-dependent Schrödinger equation describes the properties of molecular systems with high accuracy, but anything more complex than the equilibrium state of a few atoms cannot be handled at this ab initio level. Quantum theory as a model for describing materials behavior is valid also on the macroscopic scale, but the application of the (non-relativistic) Schrödinger equation to many particle systems of macroscopic size is completely in vain due to the non-tractable complexity of the involved calculations. Hence, approximations are necessary; the larger the complexity of a system and the longer the involved time span of the investigated processes are, the more severe the required approximations are. For example, at some point, the ab initio approach has to be abandoned completely and replaced by empirical parameterizations of the used model. Therefore, depending on the kind of question that one asks and depending on the desired accuracy with which specific structural features of the considered system are resolved, one has the choice between many different models which often can be usefully employed on a whole span of length and time scales.

^{−15}m) [36], at the Ångstrøm scale (10

^{−10}m) of solid state crystal lattices [37], or at the micrometer scale (10

^{−6}m), simulating grain growth processes of polycrystal solid states [38]. Thus, before getting started with computer simulations, as always in research, it is important to establish first, which phenomena and properties one is primarily interested in and which questions one is going to ask.

#### 2.1. Computer Simulations as a Research Tool

^{23}) particles, atoms, molecules or abstract constituents of a system.

## 3. Simulation Methods for Different Length and Time Scales

- static equilibrium properties, e.g., the radial distribution function of a liquid, the potential energy of a system averaged over many timesteps, the static structure function of a complex molecule, or the binding energy of an enzyme attached to a biological lipid membrane.
- dynamic or non-equilibrium properties, such as diffusion processes in biomembranes, the viscosity of a liquid, or the dynamics of the propagation of cracks and defects in crystalline materials.

#### 3.1. Electronic/Atomistic Scale

#### The Born-Oppenheimer Approximation

_{c}/m

_{e}≫ 10

^{3}), the electrons are able to follow almost instantaneously the only slowly occurring change in the core positions. Thus, the electrons are assumed to always be in the ground state associated to the actual position of the nuclei. This is the reason why the degrees of freedom of the atom cores and the electrons can be separated (Born-Oppenheimer Approximation) [99].

_{1}, ..., dV

_{N}

_{+}

_{K}of configuration space centered at the point ( $\overrightarrow{R},\overrightarrow{r}$) as:

_{n}is the nuclear wave function, and where the electronic wave function ${\phi}_{n}(\overrightarrow{R},\overrightarrow{r})$ does not depend on time anymore but only on the nuclear coordinates $\overrightarrow{R}$. Using a Taylor expansion of the stationary Schrödinger equation and several approximations that rely on the difference in masses between electrons and nuclei, see for example Chapter 8 in [100], the stationary Schrödinger equation can be separated into two equations, the electronic Schrödinger equation and an equation for the nuclei. The first equation describes how the electrons behave when the position of the nuclei is fixed. Its solution leads to an effective potential that appears in the equation for the nuclei and describes the effect of the electrons on the interaction between the nuclei. Thus, the cores move within an energy landscape of the surrounding, fast moving electrons. After restriction to the ground state φ

_{0}and further approximations (neglecting all coupling terms) [101], one obtains a classical-mechanical model for the core movements determined by the force [101]

_{0}associated with the ground state energy E

_{0}the electronic Hamilton operator ${\mathcal{H}}_{e}$ fulfills the eigenvalue equation

#### Car-Parinello MD

_{i}of the orbitals {Ψ

_{i}}. When ${\mathcal{H}}_{e}$ is diagonalized in each timestep ( ${\mathcal{H}}_{e}$ Ψ

_{i}= ε

_{i}Ψ

_{i}with ε

_{i}= Λ

_{ij}) the classical forces acting on the nuclei can be calculated and integrated according to the Newtonian equations of motion for the degrees of freedom of the nuclei:

_{i}}. Thus, CP generates a classical dynamics of the nuclei in phase space while the dynamics of the electrons is purely fictitious (only the solution of the time dependent Schrödinger equation generates the correct electronic motion). The self-consistent solution of the electronic problem is avoided and substituted in each MD timestep by a dynamic propagation of the orbitals, which are considered as classical fields with constraints. We note that the CP method in the limit of zero orbital masses μ

_{i}yields the Born Oppenheimer result, so it is a controlled approximation for Born Oppenheimer dynamics.

#### 3.2. Atomistic/Microscopic Scale

^{−12}s to 10

^{−6}s for the longest runs on the largest supercomputers.

#### 3.3. Microscopic/Mesoscopic Scale

^{−8}s to 10

^{−4}s. This is the typical domain of soft matter and biological systems, e.g., polymers, amphiphiles or colloidal systems. It is the scale on which self-organization of matter in biological systems, e.g., cells or membranes, occurs. These systems are driven by an interplay between their energy and entropy, as there are many configurations and packing densities available to these systems.

#### 3.4. Mesoscopic/Macroscopic Scale

## 4. The Key Ingredients of Molecular Dynamics Simulations

^{−15}s). With several million timesteps that are usually simulated in a MD run, the largest available length- and timescales for atomic systems are typically limited to the order of a few hundred nanometers simulated for a few hundred nanoseconds [113,114]. With larger computer systems, one will be able to simulate even larger systems; however, the available time scale does not necessarily grow with the number of available processors, as the time domain cannot be decomposed distributed over many CPUs as it is done when decomposing the spatial domain.

^{10}particles [113], opening the route to investigations of physical structure-property phenomena on the micrometer scale, which overlaps with typical computational methods based on continuum theory. Figure 8 exhibits the necessary computer hardware necessary in modern computational science along with a photograph of the first electronic computer, the ENIAC, developed at the Los Alamos Laboratories and which started to operate in 1946.

- The knowledge of one single structure, even if it is the structure of the global energy minimum, is not sufficient. It is always necessary to generate a representative ensemble at a given temperature, in order to compute macroscopic properties.
- The atomic details of structure and motion obtained in molecular simulations, is often not relevant for macroscopic properties. This opens the route for simplifications in the description of interactions and averaging over irrelevant details. Statistical mechanics provides the theoretical framework for such simplifications.

_{i}and position vectors ${\overrightarrow{r}}_{i}$ may be written as

#### 4.1. Limitations of MD

- Artificial boundary conditionsThe system size that can be simulated with MD is very small compared to real molecular systems. Hence, a system of particles will have many unwanted artificial boundaries (surfaces). In order to avoid real boundaries one introduces periodic boundary conditions (see Section 4.3.) which can introduce artificial spatial correlations in too small systems. Therefore, one should always check the influence of system size on results.
- Cut off of long-range interactionsUsually, all non-bonded interactions are cut-off at a certain distance in order to keep the cost of force computation (and the search effort for interacting particles) as small as possible. Due to the minimum image convention (see Section 4.4.) the cutoff range may not exceed half the box size. While this is large enough for most systems in practice, problems are only to be expected with systems containing charged particles. Here, simulations can go wrong badly and, e.g., lead to an accumulation of the charged particles in one corner of the box. Here, one has to use special algorithms such as the particle-mesh Ewald method [115,116].
- The simulations are classicalUsing Newton’s equations of motion implies the use of classical mechanics for the description of the atomic motion. All those material properties connected with the fast electronic degrees of freedom are not correctly described. For example, atomic oscillations (e.g., covalent C-C-bond oscillations in polyethylene molecules, or hydrogen-bonded motion in biopolymers such as DNA, proteins or biomembranes) are typically of the order 10
^{14}Hz. The specific heat is another example which is not correctly described in a classical model as here, at room temperature, all degrees of freedom are excited, whereas quantum mechanically, the high-frequency bonding oscillations are not excited, thus leading to a smaller (correct) value of the specific heat than in the classical picture. A general solution to this problem is to treat the bond distances and bond angles as constraints in the equations of motion. Thus, the highest frequencies in the molecular motion are removed and one can use a much higher timestep in the integration [117]. - The electrons are in the ground stateUsing conservative force fields in MD implies that the potential is a function of the atomic positions only. No electronic motions are considered, thus the electrons remain in their ground state and are considered to follow the core movements instantaneously. This means that electronically excited states, electronic transfer processes and chemical reactions cannot be treated.
- Approximative force fieldsForce fields are not really an integral part of the simulation method but are determined from experiments or from a parameterization using ab initio methods. Also, most often, force fields are pair-additive (except for the long-range Coulomb force) and hence cannot incorporate polarizabilities of molecules. However, such force fields exist and there is continuous effort to generate such kind of force fields [118,119]. In most practical applications however, e.g., for biomacromolecules in aqueous solution, pair potentials are quite accurate mostly because of error cancellation. This does not always work, for example ab initio predictions of small proteins still yields mixed results and when the proteins fail to fold, it is often unclear whether the failure is due to a deficiency in the underlying force fields or simply a lack of sufficient simulation time [120,121].
- Force fields are pair additiveAll non-bonded forces result from the sum of non-bonded pair interactions. Non pair-additive interactions such as the polarizability of molecules and atoms, are represented by averaged effective pair potentials. Hence, the pair interactions are not valid for situations that differ considerably from the test systems on which the models were parameterized. The omission of polarizability in the potential implies that the electrons do not provide a dielectric constant with the consequence that the long-range electrostatic interaction between charges is not reduced (as it should be) and thus overestimated in simulations.

#### 4.2. Molecular Interactions

#### Non-bonded Interactions

_{nb}, i.e., the potential hypersurface of the non-bonded interactions can be written as [7]

_{1}, φ

_{2}, φ

_{3}, ... are the interaction contributions due to external fields (e.g., the effect of container walls) and due to pair, triple and higher order interactions of particles. In classical MD one often simplifies the potential by the hypotheses that all interactions can be described by pairwise additive potentials. Despite this reduction of complexity, the efficiency of a MD algorithm taking into account only pair interactions of particles is rather low (of order $\mathcal{O}({N}^{2})$) and several optimization techniques are needed in order to improve the runtime behavior to $\mathcal{O}(N)$.

_{i}– r

_{j}| is a potential of the following form:

_{1}and C

_{2}are parameters of the attractive and repulsive interaction and the electrostatic energy Φ

_{Coulomb}(r) between the particles with position vectors ${\overrightarrow{r}}_{i}$ and ${\overrightarrow{r}}_{j}$ is given by:

_{air}= 1 for air, ε

_{prot}= 4 for proteins or ε

_{H20}= 82 for water. The z

_{i}denote the charge of individual monomers in the macromolecule and e is the electric charge of an electron.

_{2}O-molecule. If the electronegativity of one atom is large enough, it can attract the whole electron from the bonding partner. This is the case for example with NaCl where the initially electric neutral Cl atom becomes a Cl

^{−}-ion and Na turns into Na

^{+}accordingly. Chemical bonds which emerge from Coulomb attraction of ions are called ionic bonds. This type of chemical bond plays an important role for the formation of structures of biomolecules. For example, charged sidegroups may bind to receptors within the cell membrane or protein structures are be stabilized when a positively charged, protonated ammonium group (

^{+}NH

_{4}) and a negatively charged carboxyl group (COOH

^{−}) form an ionic bonding.

^{3}term in the potential energy between two dipoles with moments ${\overrightarrow{p}}_{1}$ and ${\overrightarrow{p}}_{2}$.

_{p}of a dipole which is induced by an electric field $\overrightarrow{E}$ one obtains by elementary integration:

^{6}and consequently rather weak. The polarizability of water molecules gives rise to another, directed interaction which is called hydrogen bond. Hydrogen bonds are not only found in fluid and solid water but also in complex biopolymers and macromolecules, for example in proteins, where hydrogen bonds are responsible for the genesis of tertiary structures such as α-helices or β-sheets. Despite the directed nature of the hydrogen bond one often assumes a spherically symmetric analytic form of the type (A·r

^{−}

^{12}– B·r

^{−}

^{6}), but also a more precise form taking into account the non-linearity of the hydrogen bond by angle θ between N-H-O have been proposed [131]:

_{0}.

_{0}the length scale. In simulations one uses dimensionless reduced units which tend to avoid numerical errors when processing very small numbers, arising e.g., from physical constants such as the Boltzmann constant k

_{B}= 1.38·10

^{−23}J/K. In these reduced (simulation) units, one MD timestep is measured in units of τ̂ = (mσ

^{2}/ε)

^{1/2}, where m is the mass of a particle and ε and σ

_{0}are often simply set to σ

_{0}= ε = k

_{B}T = 1. Applied to real molecules, for example to Argon with m = 6.63 × 10

^{−23}kg, σ

_{0}≈ 3.4 × 10

^{−10}m and ε/k

_{B}≈120K one obtains a typical MD timestep τ̂ ≈ 3.1 × 10

^{−13}s.

^{−}

^{12}term, one obtains the Buckingham potential [133]:

_{cut}smooth. The parameters α, β and γ are determined analytically such that the potential tail of Φ

_{cos}has zero derivative at r = 2

^{1/6}and at r = r

_{cut}, while it is zero at r = r

_{cut}and has value γ at r = 2

^{1/6}, where γ is the depth of the attractive part. Further details can be found in [51]. When setting r

_{cut}= 1.5 one sets γ = −1 and obtains α and β as solutions of the linear set of equations

#### Bonded Interactions

_{i}, bond angles θ and torsion angles φ. When neglecting the fast electronic degrees of freedom, often bond angles and bond lengths can be assumed to be constants. In this case, the potential includes lengths l

_{0}and angles θ

_{0}, φ

_{0}at equilibrium about which the molecules are allowed to oscillate, and restoring forces which ensure that the system attains these equilibrium values on average. Hence the bonded interactions Φ

_{bonded}for polymeric macromolecular systems with internal degrees of freedom can be treated by using some or all parts of the following potential term:

_{0}, θ

_{0}and φ

_{0}are the equilibrium distance, bond angle and torsion angle, respectively.

_{0}[51]:

#### 4.3. Periodic Boundary Conditions

#### 4.4. Minimum Image Convention

_{B}with box length L

_{B}. This procedure is called minimum image convention. Using the minimum image convention, each particle interacts with at the most (N – 1) particles. Particularly for ionic systems a cut-off has to be chosen such that the electro-neutrality is not violated. Otherwise, particles would start interacting with their periodic images which would render all calculations of forces and energies erroneous.

#### 4.5. Force Calculation

^{2}efficiency. This algorithm becomes extremely inefficient for systems of more than a few thousand particles, cf. Figure 12a.

#### Linked-Cell Algorithm

_{cut}for the potential. The idea here is to neglect all contributions in the sums in Equations (28) and (29) that are smaller than the threshold r

_{cut}which characterizes the range of the interaction. Thus, in this case the force ${\overrightarrow{F}}_{i}$ on particle i is approximated by

_{cut}are neglected. This introduces a small error in the computation of the forces and the total energy of the system, but it reduces the overall computational effort from $\mathcal{O}({N}^{2})$ to $\mathcal{O}(N)$. For systems with short-ranged or rapidly decaying potentials, a very efficient algorithm for the search of potentially interacting particles, i.e., those particles that are within the cutoff distance r

_{cut}of a particle i, has been developed [4]. In MD this algorithm can be implemented most efficiently by geometrically dividing the volume of the (usually cubic) simulation box into small cubic cells whose sizes are slightly larger than the interaction range r

_{cut}of particles, cf. Figure 12b. The particles are then sorted into these cells using the linked-cell algorithm (LCA). The LCA owes its name to the way in which the particle data are arranged in computer memory, namely as linked list for each cell. For the calculation of the interactions it is then sufficient to calculate the distances between particles in neighboring cells only, since cells which are further than one cell apart are by construction beyond the interaction range. Thus, the number of distance calculations is restricted to those particle pairs of neighboring cells only which means that the sums in Equation (30) are now split into partial sums corresponding to the decomposition of the simulation domain into cells. For the force ${\overrightarrow{F}}_{i}$ on particle i in cell number n one obtains a sum of the form

_{cut}, cf. Figure 12. For simulations of dense melts with many particles, this requirement is usually met. Consequently, by this method, the search-loop effort is reduced to $\mathcal{O}(N)$, but with a pre-factor that still can be very large, depending on the density of particles 〈ρ〉 and the interaction range r

_{cut}.

#### Linked-Cell Algorithm With Neighbor-Lists

_{cut}surrounding a specific particle i actually interact with each this particle, cf. Figure 12c. In order to speed up the search algorithm for identifying interacting particles, the volume between the outer sphere of radius (r

_{cut}+ r

_{skin}) and the smaller one is additionally covered by the neighbor list for particle i. Thus, this list contains not only actually interacting particles at some specific point in time, but it also contains all particles that might enter the interaction range of the inner sphere within the next few timesteps. This greatly speeds up the simulation, because the list of potentially interacting particles will be valid for several timesteps, in the order of 5–15, before it has to be rebuilt. The interval, at which list-reconstruction has to be done, depends upon r

_{cut}, the particle density ρ and the skin radius r

_{skin}. Once a particle has moved a distance larger than ${d}^{2}={\left(\frac{{r}_{\text{skin}}}{2}\right)}^{2}$, the update is due. The accumulated distance that each particle moved can be readily monitored during the distance calculation. Tests of this method with the bonded potential of Equation (25) and the FENE potential of Equation (27) for flexible macromolecules with r

_{cut}= 1.5σ and ρ = 0.85σ show that a radius of r

_{skin}≈ 0.35σ to 0.40σ is the optimal choice [3,51]. The d

_{i}length of the cells in each direction is given by L

_{i}/modulo ( ${L}_{i}/{r}_{\text{cut}}^{\text{max}}$) with L

_{i}being the box size in each direction and ${r}_{\text{cut}}^{\text{max}}$ being the largest cutoff of all potentials that are used. The cells are numbered, beginning with the one in the lower left corner of the simulation box where the origin of the coordinate system is located. Each time when an update of the neighbor list is due, the particles are periodically back-folded into the simulation box and then sorted into the different cells according to their coordinates. Subsequently, only the distances of particles of neighbor cells are calculated, with each cell having 26 neighbors in three dimensions. Again, due to Newton’s third axiom only half of them have to be considered.

#### Ghostparticles

_{x}× L

_{y}× L

_{z}.. This periodic wrap-around is done in the innermost loop of the force calculation and therefore is extremely expensive in terms of simulation time.

^{3}particles, on average ≈ 73% of all particles are ghosts, whereas this number has decreased to an average value of ≈ 13% for a system with the same density, but N = 2 × 10

^{5}. Using this technique for larger system can result in a overall speed-up of up to a factor of 2.

#### 4.6. Efficiency of the MD Method

- Testing an if-condition,
- Assigning a value, i.e., changing the contents of a memory,
- Executing one of the elementary operations (+, −, ×, DIV, MOD),
- Initializing a loop variable.

- START
- for i := 1 TO N - 1 DO
- for j := 1 TO N DO
- if a[i] > a[j] then h = a[i]; a[i] = a[j]; a[j] = h
- END

^{−9}seconds, one can sort arrays containing 2 × 10

^{5}elements within one second. Often however, one is only interested in how the run time of an algorithm depends on the number of input elements N, only considering the leading term in the computation time. In the example above one would speak of a “quadratic”, or “order N

^{2}” runtime and write symbolically $\mathcal{O}({N}^{2})$. The meaning of this symbolic $\mathcal{O}$-notation is the following:

_{0}such that for all N ≥ N

_{0}: g(N) ≤ c × f (N). For example, the function 3N

^{2}+ 4N is of order N

^{2}, or in symbolic notation: 3N

^{2}+ 4N = $\mathcal{O}({N}^{2})$, as one can choose c = 3. Then 3N

^{2}+ 4N ≤ 3N

^{2}for all N > 5. Thus, the previous relation is true for e.g., N

_{0}= 6.

_{1}–A

_{5}with corresponding run times N, N

^{2}, N

^{3}, 2

^{N}, N!, where N is the considered system size, e.g., the number of atoms, particles, nodes or finite elements in some simulation program. We again assume that one elementary step takes 10

^{−}

^{9}seconds on a real computer.

_{4}and A

_{5}) are not acceptable for all practical purposes. For these algorithms, even with very small system sizes N one reaches run times which are larger than the estimated age of the universe (10

^{10}years). Algorithm A

_{5}could, for example, be a solution of the traveling salesman problem. If the first point out of N has been visited, there are (N – 1) choices for the second one. This finally results in an exponential run time of at the least N! steps. A runtime 2

^{N}as in A

_{4}is typical for problems where the solution space of the problem consists of a subset of a given set of N objects; There are 2

^{N}possible subsets of this basis set. The “efficient” algorithms A

_{1}, A

_{2}, A

_{3}with run times of at the most N

^{3}are the most commonly used ones in computational materials science.

_{1}, A

_{2}and A

_{3}are considered to be efficient: Assuming that the available computer systems—due to a technology jump—will be 10 or 100 times faster than today, then the efficiency of algorithms A

_{1}, A

_{2}and A

_{3}will be shifted by a factor, whereas for the exponential algorithms A

_{4}, A

_{5}the efficiency will be shifted only by an additive constant.

_{1}, A

_{2}and A

_{3}have polynomial run times. An algorithm is said to be efficient if its runtime—which depends on some input N—has a polynomial upper bound. For example, the runtime function $2{N}^{4}{({\text{log}}_{2}N)}^{4}+3\sqrt{N}$ has a polynomial upper bound (for large N), e.g., N

^{5}. In $\mathcal{O}$-notation this is expressed as $\mathcal{O}({N}^{k})$ with k being the degree of the polynomial. Algorithms A

_{4}and A

_{5}on the other hand have no polynomial upper limit. Thus, they are called inefficient. In computer science, the class of problems that can be solved with efficient algorithms (i.e., algorithms that are polynomially bounded) are denoted with the letter

**P**, cf. Figure 14a. As the set of polynomials is closed under addition, multiplication and composition,

**P**is a very robust class of problems: Combining several polynomial algorithms results into an algorithm which again exhibits a polynomial runtime.

**P**of efficient algorithms, an inefficient algorithm can have a shorter runtime than its efficient counterpart, up to a certain system size N

_{0}. For example, an algorithm with a runtime 1000 × N

^{1000}falls into the class

**P**whereas an algorithm with a runtime 1.1

^{N}is exponential and thus inefficient. However, the exponential algorithm only exhibits longer runtime than the efficient one for system sizes up to N ∼ 123, 000, cf. Figure 14b.

^{2}– 1 ES. For a randomly shuffled array one can show that the expectation value for the number of elementary steps is ${\sum}_{k=2}^{N}(1/k)\approx \text{ln}\hspace{0.17em}N$ [138]. Thus, with a randomly sorted array the total number of ES in this example is roughly N

^{2}+ 3N ln N. Hence, the actual runtime of an algorithm lies somewhere between the worst-case and the average-case runtime behavior, cf. Figure 14c.

^{3}or even higher polynomial dependency of the run time. This is the main reason why ab initio methods are restricted to very small system sizes.

#### Amdahl’s Law

_{1}the execution time for a sequential program. If a fraction f of this program can be parallelized using M processors, then the theoretical execution time is determined by the sum of the time T

_{s}= (1 – f)T

_{1}which is needed for the serial part and the time T

_{p}= (f·T

_{1})/M needed for the parallelized program part. The maximum speedup S (f, M) of a parallelized code is thus given by:

## 5. Application: Simulating the Effect of Shock Waves in Polycrystalline Solid States

_{S}one can relate the conditions ahead and behind the shock to each other via the conservation equations for mass, momentum and energy. With the thermodynamic conditions described by the mass density ρ, the pressure p and the specific internal energy e, and using index 0 for the initial and 1 for the shocked state, respectively, the Rankine-Hugoniot equations describe the jump conditions to be:

_{S}along with the particle velocity υ

_{1}. Thus, the measured relation between shock velocity and particle velocity

_{ref}is given by the Hugoniot curve and the Grüneisen coefficient is $\Gamma =V{\frac{\partial p}{\partial e}|}_{V}$.

#### 5.1. Modeling Polycrystalline Solids Using Power Diagrams

^{2}with convex polyhedra, i.e., as a polyhedral cell complex, cf. Figure 16a. A direct, primitive discretization of the micro-photograph into equal-spaced squares in a 2D mesh can be used for a direct simulation of material properties, cf. Figure 16b. However, with this modeling approach, the grain boundaries on the micrometer scale have to be modeled explicitly with very small elements of finite thickness. Thus, the influence of the area of the interface is unrealistically overestimated in light of the known fact that grain boundaries, which constitute an area of local disorder, often exhibit only a thickness of a few layers of atoms [154]. Moreover, a photomicrograph is just one 2D sample of the real micro structure in 3D, hence the value of its explicit rendering is very questionable. Finally, with this approach there is no 3D information available at all. While experimentally measured micro structures in 3D are generally not available for ceramic materials, only recently first reports about measured micro structures of steel have been published [155,156]. Nevertheless, these experiments are expensive and their resolution as well as the number of measured grains still seem to be poor [156].

^{2}have been used in many fields of materials science, e. g. for the description of biological tissues or polymer foams [161]. Ghosh et al. [162] utilized Voronoi cells to obtain stereologic information for the different morphologies of grains in ceramics and Espinoza et al. [163] used random Voronoi tessellations for the study of wave propagation models that describe various mechanisms of dynamic material failure at the micro scale. However, these models have major drawbacks such as limitations to two dimensions and a generic nature of the structures as they are usually not validated with actual experimental data. Besides its applications in other fields of science, the Voronoi diagram and its dual can be used for solving numerous, and surprisingly different, geometric problems. Moreover, these structures are very appealing, and a lot of research has been devoted to their study (about one in every 16 papers in computational geometry), ever since Shamos and Hoey [164] introduced them to this field. The reader interested in a complete overview over the existing literature should consult the book by Okabe et al. [165] who list more than 600 papers, and the survey by Aurenhammer [166].

_{i}

^{exp}and P

_{i}

^{exp}by calculating the first k central moments of the area and perimeter distributions A

_{i}and P

_{i}, respectively. A figure of merit m of conformity is defined according to which the PDs are optimized [168]:

^{−1}. If m = 0 is reached, the first k central moments of the experimental distributions agree completely with the model. In Figure 20 we present the resulting histogram of an optimized PD for Al

_{2}O

_{3}and show the time development of the figure of merit m for this sample, following the proposed optimization scheme described above.

#### 5.2. A Particle Model for Simulating Shock Wave Failure in Solids

^{8}particles during the late 1990s by Abraham and Coworkers [33,34], Holian and Lomdahl [189], Zhou [175] and others [190,191]. Today, many-particle MD simulations taking into account the degrees of freedom of several billion particles have been simulated in atomistic shock wave and brittle to ductile failure simulations [192–194].

#### Model Potentials

^{ij}decreases compared to the initial overlap d

_{0}

^{ij}:

_{0}

^{3}ensures the correct scaling behavior of the calculated total stress ∑

_{ij}σ

^{ij}= ∑

_{ij}F

^{ij}/A which, as a result, is independent of N. Figure 26 shows that systems with all parameters kept constant, but only N varied, lead to the same slope (Young’s modulus) in a stress-strain diagram. In Equation (47)R

_{0}is the constant radius of the particles, d

^{ij}= d

^{ij}(t) is the instantaneous mutual distance of each interacting pairs {ij} of particles, and d

_{0}

^{ij}= d

^{ij}(t = 0) is the initial separation which the pair {ij} had in the starting configuration. Every single pair {ij} of overlapping particles is associated with a different initial separation d

_{0}

^{ij}and hence with a different force. The minimum of each individual particle pair {ij} is chosen such that the body is force-free at the start of the simulation.

_{0}

^{ij}. This property is expressed in the cohesive potential by the following equation:

_{0}again ensures a proper intrinsic scaling behavior of the material response. The total potential is the following sum:

_{tot}acts only on particle pairs that are closer together than their mutual initial distance d

_{0}

^{ij}, whereas the harmonic potential Φ

_{coh}either acts repulsively or cohesively, depending on the actual distance d

^{ij}. Failure is included in the model by introducing two breaking thresholds for the springs with respect to compressive and to tensile failure, respectively. If either of these thresholds is exceeded, the respective spring is considered to be broken and is removed from the system. A tensile failure criterium is reached when the overlap between two particles vanishes, i.e., when:

_{0}and overall particle density Ω are reached. From the initial particle pair distance distribution 〈d

_{0}

^{ij}〉 one can derive a maximal expectation value for σ

_{max}[195]:

_{0}

^{ij}〉 is to 2R

_{0}, the less is the maximum tensile strength. The random distribution of initial particle distances ultimately determines the system’s stability upon load, as well as its failure behavior and the initiation of cracks.

#### Strain, Shear and Impact Load

_{2}O

_{3}) in a quasistatic uniaxial tensile load simulation. Then, without any further fitting of parameters, the system is sheared and an impact experiment, as described in Figures 22 and 23 is performed.

_{2}O

_{3}, Ω = 1.1, ρ = 3.96 and κ = 350 are chosen; these values corresponding to a typical experimental situation of 99% volume density, ρ = 3.96 g/cm

^{3}and E = 370 GPa. Results of the simulations are displayed in the picture series of Figure 28, which shows 4 snapshots of the fracture process, in which the main features of crack instability, as pointed out by Sharon and Fineberg [184] (onset of branching at crack tips, followed by crack branching and fragmentation) are well captured. At first, many micro-cracks are initiated all over the material by failing particle pair bonds. These micro-cracks lead to local accumulations of stresses in the material until crack tips occur where local tensions accumulate to form a macroscopic crack. This crack ultimately leads to a macroscopic, catastrophic failure of the model solid, which corresponds very well to the fracture pattern of a brittle material. Similar FEM simulations, see Figure 29, using about 50 million elements, still exhibit a strong dependence of the number of elements and of element size [202]. One advantage of the proposed particle model in contrast to FEM models is, that many systems with the same set of parameters (and hence the same macroscopic properties) but statistically varying micro-structure (i.e., initial arrangement of particles) can be simulated, which is very awkward to attain using FEM. By way of performing many simulations a good statistics for corresponding observables can be achieved.

_{2}O

_{3}and SiC). These oxide and non-oxide ceramics represent two major classes of ceramics that have many important applications. The impactor hits the target at the left edge. This leads to a local compression of the particles in the impact area. The top series of snapshots in Figure 31a shows the propagation of a shock wave through the material. The shape of the shock front and also the distance traveled by it correspond very well to the high-speed photographs in the middle of Figure 31a. These snapshots were taken at comparable times after the impact had occurred in the experiment and in the simulation, respectively. In the experiments which are used for comparison, specimens of dimensions (100 × 100 × 10)mm were impacted by a cylindrical blunt steel projectile of length 23mm, mass m = 126 g and a diameter of 29.96mm [18]. After a reflection of the pressure wave at the free end of the material sample, and its propagation back into the material, the energy stored in the shock wave front finally disperses in the material. One can study in great detail the physics of shock waves traversing the material and easily identify strained or compressed regions by plotting the potential energies of the individual pair bonds. Also failure in the material can conveniently be visualized by plotting only the failed bonds as a function of time, cf. the bottom series of snapshots in Figure 31a. A simple measure of the degree of damage is the number of broken bonds with respect to the their total initial number. This quantity is calculated from impact simulations of Al

_{2}O

_{3}and SiC, after previously adjusting the simulation parameters γ, λ and α accordingly. Figure 31b exhibits the results of this analysis. For all impact speeds the damage in the SiC-model is consistently larger than in the one for Al

_{2}O

_{3}which is also seen in the experiments.

## 6. Coarse-Grained MD Simulations of Soft Matter: Polymers and Biomacromolecules

#### 6.1. Coarse-Grained Polymers

_{g}) obeys a relation X ∝ N

^{k}, where N ∈ ℕ is the size of the system, and k ∈ ℚ is the fractal dimension which is of the form k = p/q with p ≠ = q, q ∈ ℕ and p ε ℕ. The basic properties that are sufficient to extract many structural static and dynamic features of polymers are:

- The connectivity of monomers in a chain.
- The topological constraints, e.g., the impenetrability of chain bonds.
- The Flexibility or stiffness of monomer segments.

#### 6.2. Scaling of Linear, Branched and Semiflexible Macromolecules

_{0}. This beam becomes scattered by a sample and the intensity I of the scattered waves is registered by a detector (D) at a distance d, under variation of the direction of observation. The scattering vector $\overrightarrow{q}$ is defined as

_{B}by

_{m}represents the total number of particles (or monomers in the case of macromolecules) in the sample and I

_{m}is the scattering intensity produced by one particle, if placed in the same incident beam. The scattering function expresses the ration between the actual intensity which would be measured and the intensity which would be measured, if all particles in the sample were to scatter incoherently. Scattering diagrams generally emerge from the superposition and interference of the scattered waves emanating from all particles in the sample. The total scattering amplitude measured at the detector is then given by

_{i}are determined by the particle position ${\overrightarrow{R}}_{i}$ and the scattering vector $\overrightarrow{q}$ only, being given by

_{m}are exactly known at all times (within the boundaries of numerical errors when using floats in double precision as a representation of real numbers).

_{e}in the case of linear chains or the radius of gyration R

_{g}) scale with an exponent ν = 3/5. In the opposite case of a poor solvent, polymers tend to shrink and R scales with ν = 1/3. The point were the repulsive and attractive interactions just cancel each other defines the θ–point and θ–temperature, respectively. Here, the chain configuration is that of a Gaussian random coil with an exponent ν = 1/2. There are still three-body and higher order interactions present in a θ–solvent, but their contribution to the free energy is negligibly small [52]. For the description of the distance of temperature T from the θ–temperature, a dimensionless parameter is used, the reduced temperature ζ which is defined as:

_{θ},

_{θ,}

_{g}

^{2}〉 ∝ 〈R

_{e}

^{2}〉 ∝ (N – 1)

^{2νθ}with ν

_{θ}= 0.5 besides logarithmic corrections in 3D. Therefore, one expects that a plot of 〈R

^{2}〉 / (N – 1) vs. T for different values of N shows a common intersection point at T = T

_{θ}where the curvature changes: for T > T

_{θ}the larger N, the larger the ratio 〈R

^{2}〉 / (N – 1), while for T < T

_{θ}the larger N, the smaller the ratio 〈R

^{2}〉 / (N – 1). Using the model potential of Equation (25) – instead of varying temperature (which involves rescaling of the particle velocities), different solvent qualities are obtained by tuning the interaction parameter λ. The corresponding transition curves are displayed in Figure 34 which show a clear intersection point at roughly λ = λ

_{θ}≈ 0.65. Moreover it can be seen that the transition becomes sharper with increasing chain length N. The different curves do not intersect exactly at one single point, but there is an extended region in which the chains behave in a Gaussian manner. The size of this region is ∝ N

^{−1/2}[52]. There is a very slight drift of the intersection point toward a smaller value of λ with increasing chain length N.

^{−1/2}therefore yields different straight lines that intersect each other exactly at T = T

_{θ}and λ = λ

_{θ}respectively. This extrapolation (N → ∞) is displayed in Figure 35. The different lines do not intersect exactly at N

^{−1/2}= 0 which is due to the finite length of the chains. As a result of these plots one yields the value of λ for which the repulsive and attractive interactions in the used model just cancel each other:

_{b}is the (constant) bondlength of the monomers (often also called segment length). The importance of S(q) lies in the fact that it is directly measurable in scattering experiments. For ideal linear chains the function S(q) can be explicitly calculated and is given by the monotonously decreasing Debye function.

^{2}〈R

_{g}

^{2}〉

_{0}with index 0 denoting θ –conditions. For small values of x, corresponding to large distances between scattering units, the Debye function S(x) also provides a good description of a linear chain in a good solvent with the scaling variable x describing the expansion of the chain. For very small scattering vectors q one obtains the Guinier approximation [51] by an expansion of S(q), which is used in experiments to calculate the radius of gyration 〈R

_{g}

^{2}〉. In the intermediate regime of scattering vectors, S(q) obeys a scaling law which, in a double-logarithmic plot, should yield a slope of −1/ν. For large q-values finally, S(q) is expected to behave as 1/N. The overall expected behavior of S(q) is summarized in Equation (69).

_{θ}= 0.5. Therefore q

^{2}S(q), plotted against wave vector q, which is called a Kratky plot should approach a constant value. Figure 36 displays this behavior for different chain lengths with high resolution in terms of λ. The respective dotted horizontal line is a guide to the eye. The larger the chains are, the smaller is the λ-range at which the chains display ideal (Gaussian) random–walk behavior. For large values of λ the chains are collapsed and form compact globules the local structure of which is also reflected in the structure function by several distinct peaks for larger q-values. These peaks become the more pronounced the longer the chains are, reflecting the fact that the transition curves become ever sharper with increasing chain length. Hence, longer chains are already in the collapsed regime for values of λ at which the smaller chains still exhibit Gaussian behavior. The structure function of the largest system in Figure 36 for λ = 1.0 already resembles very much the scattering pattern of a sphere.

_{g}

^{2}〉 for different star polymers as a function of N and for different functionalities f is displayed. Functionality f = 2 corresponds to linear chains, f = 3 corresponds three-arm star polymers and so on. The star polymers were generated with the MD simulation package “MD-Cube” developed by Steinhauser [51,63] which is capable of handling a very large array of branched polymer topologies, from star polymers to dendrimers, H-polymers, comb-polymers or randomly hyperbranched polymers. Details of the set-up of chains which works the same way for linear and branched polymer topologies can be found in [208]. Figure 37a shows a double-logarithmic plot from which one obtains the scaling exponents of R

_{g}for stars with different numbers of arms. The results for linear chains are displayed as well, for which chain lengths of up to N = 5000 were simulated. Within the errors of the simulation, the exponents do not depend on the number of arms, as expected from theory. The obtained scaling exponents are summarized in Table 4 and exhibit a reasonable agreement with theory.

^{−}

^{1}. A plot with exponents −2 or −1/2 leads to worse correlation coefficients. This result is consistent with lattice-MC simulations on a fcc-lattice [209]. More details on finite-size scaling can be found in [50,51].

_{p}(or Kuhn segment lengths l

_{K}) comparable to or larger than their contour length L and their rigidity and relaxation behavior are essential for their biological functions.

_{bend}(θ) can be introduced. Rewriting this term by introducing the unit vector ${\overrightarrow{u}}_{j}=({\overrightarrow{r}}_{j+1}-{\overrightarrow{r}}_{j})/|{\overrightarrow{r}}_{j+1}-{\overrightarrow{r}}_{j}|$ along the macromolecule, cf. Figure 10, one obtains:

_{i}is the angle between ${\overrightarrow{u}}_{j}$ and ${\overrightarrow{u}}_{j+1}$. The crossover scaling from coil-like, flexible structures on large length scales to stretched conformations at smaller scales can be seen in the scaling of S(q) when performing simulations with different values of k

_{θ}[208]. Results for linear chains of length N = 700 are displayed in Figure 38a. The chains show a scaling according to q

^{ν}. The stiffest chains exhibit a q

^{−}

^{1}–scaling which is characteristic for stiff rods. Thus, by varying parameter k

_{θ}, the whole range of bending stiffness of chains from fully flexible chains to stiff rods can be covered. The range of q–vectors for which the crossover from flexible to semiflexible and stiff occurs shifts to smaller scatter vectors with increasing stiffness k

_{θ}of the chains. The scaling plot in Figure 38b shows that the transition occurs for q ≈ 1/l

_{K}, i.e., on a length scale of the order of the statistical Kuhn length. In essence, only the fully flexible chains (red data points) exhibit a deviation from the master curve on large length scales (i.e., small q–values), which corresponds to their different global structure compared with semi-flexible macromolecules. Examples for snapshots of stiff and semiflexible chains are finally displayed in Figure 39.

#### 6.3. Polyelectrolytes

_{B}is given by:

_{B}is the Boltzmann constant, T is temperature, epsilon is the energy scale from the Lennard-Jones potential of Equation 21, σ defines the length scale (size of one monomer) and e is the electronic charge. The interaction parameter for the presented study is chosen in the range of ξ = 0, ..., 100 which covers electrically neutral chains (ξ = 0) in good solvent as well as highly charged chain systems (ξ = 100). The monomers in the chains are connected by harmonic bonds using the first term of the bonded potential of Equation (26). The interaction with the solvent is taken into account by a stochastic force ( ${\overrightarrow{\Gamma}}_{i}$) and a friction force with a damping constant χ, acting on each mass point. The equations of motion of the system are thus given by the Langevin equations

_{cut}= 1.5, Equation 11 with k = 1 and z

_{i/j}= ±1, and the first term on the right-hand side of the bonded potential in Equation 26 with κ = 5000ε/σ and bond length l

_{0}= σ

_{0}= 1.0. The stochastic force ${\overrightarrow{\Gamma}}_{i}$ is assumed to be stationary, random, and Gaussian (white noise). The electrically neutral system is placed in a cubic simulation box and periodic boundary conditions are applied for the intermolecular Lennard-Jones interaction according to Equation (21), thereby keeping the density ρ = N/V = 2.1 × 10

^{−7}/σ

^{3}constant when changing the chain length N. The number of monomers N per chain was chosen as N = 10, 20, 40, 80 and 160 so as to cover at least one order of magnitude. For the Coulomb interaction a cutoff that is half the box length r

_{cut}= 1/2L

_{B}was chosen. This can be done as the eventually collapsed polyelectolyte complexes which are analyzed are confined to a small region in space which is much smaller than r

_{cut}. In the following we discuss exemplarily some scaling properties of charged linear macromolecules in the collapsed state. The simulations are started with two well separated and equilibrated chains in the simulation box. After turning on the Coulomb interactions with opposite charges z

_{i/j}= ±1 in the monomers of both chains, the chains start to attract each other. In a first step during the aggregation process the chains start to twist around each other and form helical like structures as presented in Figure 40. In a second step, the chains start to form a compact globular structure because of the attractive interactions between dipoles formed by oppositely charged monomers, see the snapshots in Figure 41a.

_{g}obtained for intermediate interaction strengths ξ and scaled by (N – 1)

^{2/3}. Here, the data of various chain lengths fall nicely on top of each other. This scaling corresponds to the scaling behavior of flexible chains in a bad solvent and is also in accordance with what was reported by Shrivastava and Muthukumar [226]. The change of R

_{g}is connected with a change of the density ρ of the polyelectrolyte aggregate. However, in Figure 41b, which presents an example of ρ for ξ = 4, only a slight dependence of the density on the chain length N can be observed. ρ measures the radial monomer density with respect to the center of mass of the total system. For longer chains, there is a plateau while for short chains there is a pronounced maximum of the density for small distances from the center of mass. While this maximum vanished with decreasing ξ it appears also at higher interaction strengths for longer chains. Monomers on the outer part of the polyelectrolyte complex experience a stronger attraction by the inner part of the cluster than the monomers inside of it, and for smaller ξ, chains of different lengths are deformed to different degrees which leads to a chain length dependence of the density profile.

## 7. Emerging Computational Applications in Biophysics and Medical Tumor Treatment

## 8. Concluding Remarks

## Acknowledgments

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- .

**Figure 1.**(

**a**) Micrograph section of an etched Al

_{2}O

_{3}ceramic surface exhibiting the granular structure on the microscale. (

**b**) SEM photograph of the fracture surface of Al

_{2}O

_{3}after edge-on impact experiment [18] with striking speed of v ≈ 400m/s. (

**c,d**) Microstructural details of the Al

_{2}O

_{3}surface exhibiting structural hierarchies. Figure by M.O. Steinhauser, Fraunhofer EMI.

**Figure 2.**Schematic hierarchical view of structural properties of important classes of materials contrasting typical structural features of inorganic crystalline materials (engineering materials, green) and the structural features of self-organizing organic biological materials (blue). At the nanoscale, the basic constituents of all condensed matter are the atoms bound together in chemical bonds.

**Figure 3.**(a) Light microscopic view of a glass fiber reinforced sheet moulding compound (SMC) which is used in car industry as light-weight material. (b) CT-microscopic reconstruction of a section of the laminar glass fiber structure in the material. (c) Scanning Acoustic microscopic view of a tensile-test SMC specimen exhibiting the glass fiber bundles within the compound. Figure by M.O. Steinhauser, Fraunhofer EMI.

**Figure 4.**Schematic comparing the relevant length scales in materials science according to [3]. In the field of nano- and microtechnology one usually tries to approach the molecular level from larger scales, miniaturizing technical devices, whereas nature itself always seems to follow a bottom-up approach, assembling and self-organizing its complex (soft) structures from the atomic scale to complex cellular organisms. The typical scopes of important experimental methods using microscopes is displayed as well. The validity of classical physics is limited to length scales down to approximately the size of atoms which, in classical numerical schemes, are often treated as point particles or spheres with a certain eigenvolume.

**Figure 5.**Physical, mathematical and numerical modeling scheme illustrated as flow chart. Starting from the experimental evidence one constructs physical theories for which a mathematical formulation usually leads to differential equations, integral equations, or master (rate) equations for the dynamic (i.e., time dependent) development of certain state variables within the system’s abstract state space. Analytic solutions of these equations are very rarely possible, except when introducing simplifications usually involving symmetries. Thus, efficient algorithms for the treated problem have to be found and implemented as a computer program. Execution of the code yields approximate numerical solutions to the mathematical model which describes the dynamics of the physical “real” system. Comparison of the obtained numerical results with experimental data allows for a validation of the used model and subsequent iterative improvement of the model and of theory.

**Figure 6.**Principal design of a computer simulation. Usually, some pre-processing as preparation for the main simulation is done with a pre-processor. This piece of computer code might be integrated in the main source code or – in particular in commercial codes –is written as an extra piece of code, compiled and run separately from the main simulation code. During pre-processing, many administration tasks can be done which are not related to the actual simulation run. In large-scale simulations, data are stored during execution for later analysis. This analysis and visualization is done during post-processing. The advantage of separating pre- and post-processing from the actual simulation code is that the code design remains clearly arranged and the important issue of optimization for speed only has to be done for the relevant pieces of the main simulation code. However, in large-scale simulations, involving billions of particles, even the task of data analysis can become a major problem that needs optimization and fast parallelized algorithms [41].

**Figure 7.**Design schematic of the particle-based multiscale simulation package “MD-Cube” at EMI. A kernel takes care of administrative tasks. Force modules can be added via defined interfaces as well as modules for the demands of different physical applications.

**Figure 8.**(a) The ENIAC (Electronic Numerical Integrator And Computer), one of the first electronic computers that started to operate in 1946. (The very first working electronic computer was the Z3 developed by Konrad Zuse in the 1930s in Germany). The ENIAC weight 30 tons used more than 18.000 vacuum tubes that can be seen in the picture and had a basic clock speed of 10

^{5}cycles per second. It was programmed by plugging cables and wires and setting switches using a huge plugboard that was distributed over the entire machine. US Army Photo. (b) Illustration of the available system size (edge length of a simulated cube of classical particles or atoms) and the necessary computer hardware for modern large-scale MD.

**Figure 9.**Graph of the total unbounded potential of Equation (25) which allows for modeling the effects of different solvent qualities.

**Figure 11.**Two-dimensional schematic of periodic boundary conditions. The particle trajectories in the central simulation box are copied in every direction.

**Figure 12.**MD Optimization schemes for the search of potentially interacting particles. (a) The least efficient all particle “brute force” approach with run time $\mathcal{O}({N}^{2})$ (b) The linked-cell algorithm which reduces the search effort to $\mathcal{O}(N)$. (c) The linked-cell algorithm combined with neighbor lists which further reduces the search effort by using a list of potentially interacting neighbor particles which can be used for several timesteps before it has to be updated. In this 2D representation, the radius of the larger circle is r

_{cut}+ r

_{skin}and the inner circle, which contains actually interacting particles, has radius r

_{cut}.

**Figure 13.**Schematic of the sequential construction of different ghost cell layers. An individual cell of the simulation box can be identified by the three integers (i

_{x}, i

_{y}, i

_{z}). The original box in this example has n

_{x}= n

_{z}= 5 and n

_{y}= 4 cells in each direction, i.e., there is no need for the simulation box to have cubic shape. (a) In a first step, all particles of cells with indices (i

_{x}= 1; i

_{y}= 1, ..., n

_{y}; i

_{z}= 1, ..., n

_{z}) are copied into the first layer of ghostparticles. The second layer of ghostparticles then contains all particles pertaining to cells (i

_{x}= 1, ..., n

_{x}; i

_{y}= 1; i

_{z}= 1, ..., n

_{z}) including the ghostparticles of the ghostcells from the first ghostlayer. Finally, the third layer of ghostcells is constructed by copying the particles of the cells with (i

_{x}= 1, ..., n

_{x}; i

_{y}= 1, ..., n

_{y}; i

_{z}= 1) as indicated in the figure.

**Figure 14.**(a) Venn diagram of the class

**P**(efficiently solvable problems), class

**NP**(non-deterministic polynomial, i.e., inefficiently solvable problems), and undecidable problems (orange box) for which no algorithms are known. Today, it is generally assumed that all problems in

**P**are contained in the class

**NP**, cf. Figure 14. So far, no proof that decides whether

**P**=

**NP**or

**P**≠

**NP**is known. (b) An inefficient algorithm (dashed line) can – for some small values N up to an input number N

_{0}– be more efficient than a polynomially bounded algorithm (solid line). (c) A real algorithm (dotted line) will always have a run time behavior somewhat in between the worst-case (dashed line) and best-case (solid line) run time.

**Figure 15.**Spallation failure in a semi-infinite aluminium target after impact by a 10mm aluminum sphere at 7km/s. (a) Target after impact experiment cut into half. (b) Hydrocode simulation of the impact and related shock propagation leading to the formation of spallation planes.

**Figure 16.**(a) Micrograph of a HPC (Al

_{2}O

_{3}) exhibiting the micro structure with an average grain size of 0.7 μm. (b) 2D FEM simulation of a primitive model of this micro structure with a shock impulse traveling through the material from left to right. The plane of the micrograph has been sectioned into 601 × 442 equal-spaced squares which are used as finite elements. The nodes of the upper and lower edge have been assigned $\overrightarrow{\upsilon}=0$ as boundary condition, whereas the leftmost element nodes of the sample are given an initial speed of v

_{x}= 500 m/s. The color code exhibits the pressure profile.

**Figure 17.**(a) Voronoi diagram of N=20 sites. For a finite set of generator points $\mathbb{\text{T}}$ ⊂ ${\mathbb{\text{T}}}^{d}$ the Voronoi diagram maps each p ∈$\mathbb{\text{T}}$ onto its Voronoi region R(p) consisting of all x ∈${\mathbb{\text{T}}}^{d}$ that are closer to p than to any other point in $\mathbb{\text{T}}$. (b) Delaunay triangulation for the sites in (a).

**Figure 18.**5 different Al

_{2}O

_{3}micrographs (a) and their corresponding grain statistics with respect to the grains’ perimeter (b). The right picture at the bottom of (a) and (b) exhibits a corresponding 2D virtual cut through the 3D PD, i.e., a Voronoi diagram, and its corresponding histogram. Clearly, the histograms both show no Gaussian distribution as was claimed, e.g., by Zhang et al. [167].

**Figure 19.**Optimization scheme as suggested in [168]. (a) 2D experimental photomicrograph (top) and an SEM picture of the 3D crystalline surface structure of Al

_{2}O

_{3}(bottom). (b) 2D virtual slice of a power diagram (top) and the corresponding 3D surface structure obtained from this model. (c) Comparison between 2D experimental data of (a) and the 3D model of (b).

**Figure 20.**(a) Area (top) and perimeter (bottom) distribution of one of the Al

_{2}O

_{3}micrographs of Figure 18 before (black) and after (red) optimization. The bar graphs show the respective histograms of experimental data. (b) Time development of the figure of merit m during the optimization for the structure of (a). After 358000 and 512000 optimization steps, the maximum step size of the reverse MC algorithm (changing the position of a generator point) was increased, which shows a direct influence on the speed of optimization. After 1.5 million steps the deviation between the model and experiment has dropped below 1.3 × 10

^{−4}. The inset shows the corresponding time development of m of the perimeter (red) and (area) distribution of the third central moment.

**Figure 21.**3D structures of a meshed PD. In (a) the granular surface structure, its mesh and a detailed augmented section of the mesh at the surface are displayed. (b) A different realization of a 3D structure displaying the possibility of either leaving a (more realistic) rough surface micro structure, or smoothing the surface and thus obtaining a model body with even surface [168].

**Figure 22.**Illustration of the multiscale problem. With concurrent FEM methods which include micro structural details, only a very small part of a real system can actually be simulated due to the necessary large number of elements. Figure taken from [172].

**Figure 23.**Snapshots of simulations of the edge-on impact system of Figure 22 using a primitive discretization of the geometry of the system in terms of hexahedral elements. (a) FEM with mesh resolution of 0.5mm. (b) FEM with mesh resolution of 1.0mm. (c) SPH with mesh resolution of 0.5mm. All computational results are obtained using a commercial code (Autodyn-3D) and are different in terms of the damage pattern of the cracks propagating through the material 3 μs after impact.

**Figure 24.**(a) Schematic of a crystal placed under shock loading. Initially it will compress uniaxially and then relax plastically through defects on the nanoscale, a process known as the one-dimensional to three-dimensional transition. The material may also undergo a structural transformation, represented here as a cubic to hexagonal change. The transformation occurs over a characteristic time scale. The new phase may be polycrystalline solid or melt. Once pressure is released, the microvoids that formed may grow, leading to macroscopic damage that causes the solid to fail. (b) This micrograph shows the voids that occur when a polycrystalline aluminium alloy is shocked and recovered. As the shock wave releases, the voids grow and may coalesce, resulting in material failure.

**Figure 25.**The particle Model as suggested in [195]. (a) Overlapping particles with radii R

_{0}and the initial (randomly generated) degree of overlap indicated by d

_{0}

^{ij}. Here, only two particles are displayed. In the model the number of overlapping particles is unlimited and each individual particle pair contributes to the overall pressure and tensile strength of the solid. (b) Sample initial configuration of overlapping particles (N = 2500) with the color code displaying the coordination number: red (8), yellow (6), and green (4). (c) The same system displayed as an unordered network.

**Figure 26.**(a) Schematic of the intrinsic scaling property of the proposed material model. Here, only the 2D case is shown for simplicity. The original system (Model M

_{a}) with edge length L

_{0}and particle radii R

_{0}is downscaled by a factor of 1/a into the subsystem Q

_{A}of M

_{A}(shaded area) with edge length L, while the particle radii are upscaled by factor a. As a result, model M

_{B}of size aL = L

_{0}is obtained containing much fewer particles, but representing the same macroscopic solid, since the stress-strain relation (and hence, Young’s modulus E) upon uni-axial tensile load is the same in both models. (b) Young’s modulus E of systems with different number of particles N in a stress-strain (σ–ε) diagram. In essence, E is indeed independent of N.

**Figure 27.**Sample configurations of systems with N = 10000 particles and different particle densities Ω. The color code displays the coordination numbers: blue (0), green (4), yellow (6), red (8). (a) Ω = 0.7. (b) Ω = 0.9 (c) Ω = 1.3 (d) Ω = 1.7 (e) Breaking strength σ

_{b}for different system sizes N (filled symbols) as a function of particle density Ω, compared with the theoretical breaking strength σ

_{max}(open symbols). The inset shows the stress-strain (σ – ε) relation for systems with three different initial expansion times τ. In essence, the larger the expansion time for the generation of the random initial overlap of particles, the larger is the material strength σ

_{max}.

**Figure 28.**Crack initiation and propagation in the virtual macroscopic material sample upon uni-axial tensile load using N = 10

^{4}particles. The color code is: Force-free bonds (green); Bonds under tensile load (red). (a) Initiation of local tensions. (b) Initiation of a crack tip with local tensions concentrated around this crack tip. (c) Crack propagation including crack instability. (d) Failure. (e) Averaged stress-strain (σ-ε) relation. For N = 2500 (green curve) 10 different systems were averaged, and for N = 10000 (red curve) the stress-strain relations of 5 different initial particle configurations obtained in uni-axial load simulations were averaged.

**Figure 29.**Snapshots of a 2D FEM simulation of the fracture process of a PMMA (poly-methyl methacrylate) plate which is subject to an initial uniform strain rate in vertical direction. Here, there is no statistical variability of the microstructure modeled with more than 50 × 10

^{6}elements. Thus, the model solid has to be artificially pre-notched and it still exhibits a strong dependence on the mesh size and the number of elements. Adapted from [202].

**Figure 30.**Quasi-static shear loading of a virtual material specimen with N = 2500. The color code is the same as in Figure 28 except for particle bonds under pressure which are coded in blue. (a) Onset of shear tensile bands and (orthogonal) shear pressure bands in the corners of the specimen. (b) Shear bands traversing the whole specimen. (c) Ultimate failure.

**Figure 31.**Results of a simulation of the edge-on-impact (EOI) geometry, cf. Figure 23, except this time, the whole macroscopic geometry of the experiment can be simulated while still including a microscopic resolution of the system. The impactor is not modeled explicitly, but rather its energy is transformed into kinetic energy of the particle bonds at the impact site. (a) Top row: A pressure wave propagates through the material and is reflected at the free end as a tensile wave (not shown). Middle row: The actual EOI experiment with a SiC specimen. The time interval between the high-speed photographs is comparable with the simulation snapshots above. The red arrows indicate the propagating shock wave front. Bottom row: The same simulation run but this time only the occurring damage in the material with respect to the number of broken bonds is shown. (b) Number of broken bonds displayed for different system sizes N, showing the convergence of the numerical scheme. Simulation parameters (α, γ, λ) are chosen such that the correct stress-strain relations of two different materials (Al

_{2}O

_{3}and SiC) are recovered in the simulation of uniaxial tensile load. The insets show high-speed photographs of SiC and Al

_{2}O

_{3}, respectively, 4 μs after impact.

**Figure 32.**A coarse-grained model of a polymer chain where some groups of the detailed atomic structure (yellow beads) is lumped into one coarse-grained particle (red). The individual particles are connected by springs (bead-spring model).

**Figure 33.**General set up of a scattering experiment according to [206]. Details are described in the main text.

**Figure 34.**Coil-to-globule transition from good to bad solvent behavior of a polymer chain. Plot of 〈R

_{g}

^{2}〉 / (N – 1)

^{2}

^{ν}vs. interaction parameter λ for linear chains. The points represent the simulated data and the dotted lines are guides to the eye. ν = ν

_{θ}= 0.5. Also displayed are simulation snapshots of linear chains for the three cases of a good, θ-, and a bad solvent.

**Figure 35.**Interaction parameter λ of Equation (25) vs. N

^{−1/2}for different values of the scaling function. Data points are based on the radius of gyration of linear chains [51].

**Figure 36.**Kratky plot of S(q) of linear chains (N = 2000) for different values of the interaction parameter λ.

**Figure 37.**(a) Log-Log plot of 〈R

_{g}

^{2}〉 vs. N of star polymers with different arm numbers f. For comparison, data for linear chains (f = 2) are displayed as well. (b) Scaling plot of the corrections to scaling of 〈R

_{g}

^{2}〉 (f) plotted vs. N

^{−}

^{1}in good solvent. For clarity, the smallest data point of the linear chains (f = 2, N = 50) is not displayed.

**Figure 38.**(a) S(q) of single linear chains with N = 700 and varying stiffness k

_{θ}. The scaling regimes (fully flexible and stiff rod) are indicated by a straight and dashed line, respectively. (b) Scaling plot of S(q)/l

_{K}versus q · l

_{K}using the statistical segment length l

_{K}adapted from [208].

**Figure 39.**Simulation snapshots of (a) flexible chains (k

_{θ}= 0), (b) semiflexible chains (k

_{θ}= 20), (c) stiff, rod-like chains (k

_{θ}= 50).

**Figure 40.**Twisted, DNA-like polyelectrolyte complexes formed by electrostatic attraction of two oppositely charged linear macromolecules with N = 40 at τ̂ = 0 (a), τ̂ = 10500 (b), τ̂ = 60000 (c) and τ̂ = 120000 (d), where τ̂ is given in Lennard-Jones units. The interaction strength is ξ = 8 [222,223].

**Figure 41.**(a) Radii of gyration as a function of the interaction strength ξ for various chain lengths according to [223]. The radius of gyration R

_{g}

^{2}is scaled by (N – 1)

^{2/3}, where (N –1) is the number of bonds of a single chain. Also displayed are sample snapshots of the collapsed globules with N = 40 and interaction strengths ξ = 0.4, 4, 40. (b) Radial density of monomers with respect to the center of mass of a globule and interaction strength ξ = 4 for different chain lengths, N = 20 (blue), N = 40 (red), N = 80 (green) and N = 160 (brown).

**Figure 42.**(a) Hierarchical features of collagen which determines the mechanical properties of cells, tissues, bones and many other biological systems, from atomistic to microscale. Three polypeptide strands arrange to form a triple helical tropocollagen molecule. Tropocollagen (TP) molecules assemble into collagen fibrils which mineralize by formation of hydroxipatite (HA) crystals in the gap regions which exist due to a staggered array of molecules. (b) Stress-strain response of a mineralized collagen fibril exhibiting larger strength than a non-mineralized, pure collagen fibril. Both figures adapted from [227] with permission.

**Figure 43.**(a) Schematic of the principle of ESWL. (b) Image of human body with symptomatic fibroids. Sonication pathway is superimposed on the image and the spot where irreversible thermal damage is expected is also superimposed onto this image. Adapted from [230] with permission. (c) Schematic of a magnetic resonance guided HIFU equipment for the possible treatment of tumor tissue. Figure adapted from [231] with permission.

**Figure 44.**Scheme of the disintegration of a kidney stone as a result of direct and indirect shock wave effects. Figure adapted from [235] with permission.

**Table 1.**Customary classification of length scales. Displayed are also typical scopes of different simulation methods and some typical applications pertaining to the respective scale.

Scale (m) | Typical Simulation Methods | Typical Applications |
---|---|---|

Electronic/Atomistic | ||

∼ 10^{−12} – 10^{−9} | Self-Consistent Hartree-Fock (SC-HF) [53,54] | crystal ground states, NMR, IR, UV spectra, molecular geometry, electronic properties, chemical reactions |

∼ 10^{−12} – 10^{−9} | Self-Consistent DFT [12,55,56] | |

∼ 10^{−12} – 10^{−9} | Car-Parinello (ab initio) Molecular Dynamics [13] | |

∼ 10^{−12} – 10^{−9} | Tight-Binding [57] | |

∼ 10^{−12} – 10^{−9} | Quantum Monte Carlo (QMC) [58–60] | |

Atomistic/Microscopic | ||

∼ 10^{−9} – 10^{−6} | Molecular Dynamics [45,46] | equations of state, Ising model, DNA polymers, rheology, transport properties, phase equilibrium, |

∼ 10^{−9} – 10^{−6} | Monte Carlo using classical force fields [42,44] | |

∼ 10^{−9} – 10^{−6} | Hybrid MD/MC [61–63] | |

∼ 10^{−9} – 10^{−6} | Embedded Atom Method [64–66] | |

∼ 10^{−9} – 10^{−6} | Particle in Cell [67,68] | |

Microscopic/Mesoscopic | ||

∼ 10^{−8} – 10^{−1} | MD and MC using effective force fields [51] | complex fluids, soft matter, granular matter, fracture mechanics, grain growth, phase transformations, polycrystal elasticity, polycrystal plasticity, diffusion, interface motion, dislocations, grain boundaries |

∼ 10^{−9} – 10^{−3} | Dissipative Particle Dynamics [69] | |

∼ 10^{−9} – 10^{−3} | Phase Field Models [70] | |

∼ 10^{−9} – 10^{−3} | Cellular Automata [71] | |

∼ 10^{−9} – 10^{−4} | Mean Field Theory | |

∼ 10^{−6} – 10^{2} | Finite Element Methods including microstructural features [72–75] | |

∼ 10^{−6} – 10^{2} | Smooth Particle Hydrodynamics [76,77] | |

∼ 10^{−9} – 10^{−4} | Lattice-Boltzmann [78] | |

∼ 10^{−9} – 10^{−4} | Dislocation Dynamics [79–82] | |

∼ 10^{−6} – 10^{0} | Discrete Element Method [83] | |

Mesoscopic/Macroscopic | ||

∼ 10^{−3} – 10^{2} | Hydrodynamics [84] | macroscopic flow, macroscopic elasticity, macroscopic plasticity, fracture mechanics, aging of materials, fatigue and wear |

∼ 10^{−3} – 10^{2} | Computational Fluid Dynamics [85–87] | |

∼ 10^{−6} – 10^{2} | Finite Element Methods [88–90] | |

∼ 10^{−6} – 10^{2} | Smooth Particle Hydrodynamics [8,91,92] | |

∼ 10^{−6} – 10^{2} | Finite Difference Methods [93,94] | |

∼ 10^{−6} – 10^{0} | Cluster & Percolation Models |

**Table 2.**Overview of typical run times of algorithms occurring in materials science applications. Depicted are the number of ES and the corresponding real times for the different algorithms under the assumption that one ES takes 10

^{−9}seconds.

Algorithm | run time | N = 10 | N = 20 | N = 50 | N = 100 |
---|---|---|---|---|---|

10 ES | 10 ES | 10 ES | 10 ES | ||

A_{1} | N | 10^{−8} s | 2 × 10^{−8} s | 5 × 10^{−8} s | 10^{−7} s |

100 ES | 400 ES | 2.5 × 10^{3} ES | 10^{5} ES | ||

A_{2} | N^{2} | 10^{−7} s | 4 × 10^{−7} s | 2.5 × 10^{−6} s | 10^{−5} s |

1000 ES | 8 × 10^{3} ES | 10^{5} ES | 10^{6} ES | ||

A_{3} | N^{3} | 10^{−6} s | 8 × 10^{−6} s | 10^{−4} s | 0.001 s |

1024 ES | 10^{5} ES | 10^{15} ES | 10^{30} ES | ||

A_{4} | 2^{N} | 10^{−6} s | 10^{−3} s | 13 days | ~ 10^{13} years |

~ 10^{6} ES | ~ 10^{18} ES | ~ 10^{64}ES | 10^{158} ES | ||

A_{5} | N! | 3 × 10^{−3} s | 77 years | 10^{48} years | ~ 10^{141} years |

**Table 3.**Speedup of the runtime of different algorithms assuming a hardware speedup factor of 10 and 100. The runtime of efficient polynomially bounded (class

**P**) algorithms will be shifted by a factor while exponential (class

**NP**) algorithms are only improved by an additive constant.

Algorithm | run time | efficiency | CPU speedup factor 10 | CPU speedup factor 100 |
---|---|---|---|---|

A_{1} | N | E_{1} | 10 × N_{1} | 100 × N_{1} |

A_{2} | N^{2} | E_{2} | $\sqrt{10}\times {N}_{2}=3.16\times {N}_{2}$ | $\sqrt{100}\times {N}_{2}=10\times {N}_{2}$ |

A_{3} | N^{3} | E_{3} | $\sqrt[3]{10}\times {N}_{3}=2.15\times {N}_{3}$ | $\sqrt[3]{100}\times {N}_{3}=4.64\times {N}_{3}$ |

A_{4} | 2^{N} | E_{4} | log_{2}(10 × N_{4}) = N_{4} + 3.3 | log_{2}(100 × N_{4}) = N_{4} + 6.6 |

A_{5} | N! | E_{5} | ≈ N_{5} + 1 | ≈ N_{5} + 2 |

**Table 4.**Obtained scaling exponents ν for star polymers in simulations with different arm numbers f.

f | 2 | 3 | 4 | 5 | 6 | 10 | 12 | 18 |

ν | 0.5989 | 0.601 | 0.603 | 0.614 | 0.617 | 0.603 | 0.599 | 0.601 |

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Steinhauser, M.O.; Hiermaier, S.
A Review of Computational Methods in Materials Science: Examples from Shock-Wave and Polymer Physics. *Int. J. Mol. Sci.* **2009**, *10*, 5135-5216.
https://doi.org/10.3390/ijms10125135

**AMA Style**

Steinhauser MO, Hiermaier S.
A Review of Computational Methods in Materials Science: Examples from Shock-Wave and Polymer Physics. *International Journal of Molecular Sciences*. 2009; 10(12):5135-5216.
https://doi.org/10.3390/ijms10125135

**Chicago/Turabian Style**

Steinhauser, Martin O., and Stefan Hiermaier.
2009. "A Review of Computational Methods in Materials Science: Examples from Shock-Wave and Polymer Physics" *International Journal of Molecular Sciences* 10, no. 12: 5135-5216.
https://doi.org/10.3390/ijms10125135