Review Topical Sections

Multiscale modeling, coarse-graining and shock wave computer simulations in materials science

  • Received: 02 June 2017 Accepted: 03 September 2017 Published: 13 December 2017
  • My intention in this review article is to briefly discuss several major topics of presentday computational materials science in order to show their importance for state-of-the-art materials modeling and computer simulation. The topics I discuss are multiscale modeling approaches for hierarchical systems such as biological macromolecules and related coarse-graining techniques, which provide an efficient means to investigate systems on the mesoscale, and shock wave physics which has many important and interesting multi- and interdisciplinary applications in research areas where physics, biology, chemistry, computer science, medicine and even engineering meet. In fact, recently, as a new emerging field, the use of coarse-grained approaches for the simulation of biological macromolecules such as lipids and bilayer membranes and the investigation of their interaction with shock waves has become very popular. This emerging area of research may contribute not only to an improved understanding of the microscopic details of molecular self-assembly but may also lead to enhanced medical tumor treatments which are based on the destructive effects of High Intensity Focused Ultrasound (HIFU) or shock waves when interacting with biological cells and tissue; these are treatments which have been used in medicine for many years, but which are not well understood from a fundamental physical point of view.

    Citation: Martin O. Steinhauser. Multiscale modeling, coarse-graining and shock wave computer simulations in materials science[J]. AIMS Materials Science, 2017, 4(6): 1319-1357. doi: 10.3934/matersci.2017.6.1319

    Related Papers:

    [1] Guoyi Li, Jun Wang, Kaibo Shi, Yiqian Tang . Some novel results for DNNs via relaxed Lyapunov functionals. Mathematical Modelling and Control, 2024, 4(1): 110-118. doi: 10.3934/mmc.2024010
    [2] Saravanan Shanmugam, R. Vadivel, S. Sabarathinam, P. Hammachukiattikul, Nallappan Gunasekaran . Enhancing synchronization criteria for fractional-order chaotic neural networks via intermittent control: an extended dissipativity approach. Mathematical Modelling and Control, 2025, 5(1): 31-47. doi: 10.3934/mmc.2025003
    [3] Gani Stamov, Ekaterina Gospodinova, Ivanka Stamova . Practical exponential stability with respect to hmanifolds of discontinuous delayed Cohen–Grossberg neural networks with variable impulsive perturbations. Mathematical Modelling and Control, 2021, 1(1): 26-34. doi: 10.3934/mmc.2021003
    [4] Bangxin Jiang, Yijun Lou, Jianquan Lu . Input-to-state stability of delayed systems with bounded-delay impulses. Mathematical Modelling and Control, 2022, 2(2): 44-54. doi: 10.3934/mmc.2022006
    [5] Hongwei Zheng, Yujuan Tian . Exponential stability of time-delay systems with highly nonlinear impulses involving delays. Mathematical Modelling and Control, 2025, 5(1): 103-120. doi: 10.3934/mmc.2025008
    [6] M. Haripriya, A. Manivannan, S. Dhanasekar, S. Lakshmanan . Finite-time synchronization of delayed complex dynamical networks via sampled-data controller. Mathematical Modelling and Control, 2025, 5(1): 73-84. doi: 10.3934/mmc.2025006
    [7] Shipeng Li . Impulsive control for stationary oscillation of nonlinear delay systems and applications. Mathematical Modelling and Control, 2023, 3(4): 267-277. doi: 10.3934/mmc.2023023
    [8] Qin Xu, Xiao Wang, Yicheng Liu . Emergent behavior of Cucker–Smale model with time-varying topological structures and reaction-type delays. Mathematical Modelling and Control, 2022, 2(4): 200-218. doi: 10.3934/mmc.2022020
    [9] Yanchao He, Yuzhen Bai . Finite-time stability and applications of positive switched linear delayed impulsive systems. Mathematical Modelling and Control, 2024, 4(2): 178-194. doi: 10.3934/mmc.2024016
    [10] Weisong Zhou, Kaihe Wang, Wei Zhu . Synchronization for discrete coupled fuzzy neural networks with uncertain information via observer-based impulsive control. Mathematical Modelling and Control, 2024, 4(1): 17-31. doi: 10.3934/mmc.2024003
  • My intention in this review article is to briefly discuss several major topics of presentday computational materials science in order to show their importance for state-of-the-art materials modeling and computer simulation. The topics I discuss are multiscale modeling approaches for hierarchical systems such as biological macromolecules and related coarse-graining techniques, which provide an efficient means to investigate systems on the mesoscale, and shock wave physics which has many important and interesting multi- and interdisciplinary applications in research areas where physics, biology, chemistry, computer science, medicine and even engineering meet. In fact, recently, as a new emerging field, the use of coarse-grained approaches for the simulation of biological macromolecules such as lipids and bilayer membranes and the investigation of their interaction with shock waves has become very popular. This emerging area of research may contribute not only to an improved understanding of the microscopic details of molecular self-assembly but may also lead to enhanced medical tumor treatments which are based on the destructive effects of High Intensity Focused Ultrasound (HIFU) or shock waves when interacting with biological cells and tissue; these are treatments which have been used in medicine for many years, but which are not well understood from a fundamental physical point of view.


    Since neural networks (NNs) are effective in modeling and describing nonlinear dynamics, there has been a remarkable surge in the utilization of NNs, which have contributed substantially to the fields of signal processing, image processing, combinatorial optimization, pattern recognition, and other scientific and engineering domains [1,2]. Theoretical advancements have laid a solid foundation for this progress, thereby facilitating the successful establishment of NNs as a powerful tool for a diverse range of applications. Consequently, the stability analysis of delayed neural networks (DNNs) has attracted many scholars [3,4].

    It should be noted that temporal lags are invariably encountered within NNs as a consequence of intrinsic factors, including but not limited to the finite velocity of information processing [5], which can lead to instability and degraded performance in numerous real-world applications of NNs. In this way, the prognostic capacity and resilience of the neural network would suffer severe impairment, thereby gravely undermining its efficacy and dependability. Hence, the assessment of the stability of a computing system must be conducted with great care and precision, employing accurate evaluation criteria that abide by the principles of scientific rigor to provide reliable guarantee for the system's operation. Thus, in recent years, researchers have been dedicated to analyzing the stability of DNNs and investing significant amounts of time and effort into reducing the conservatism of stability criteria [6,7,8]. In regards to the stability criterion, accommodating a wider range of delay tolerance would result in a less conservative estimate. Consequently, the upper limit of the delay range assumes critical significance in the assessment and quantification of the degree of conservativeness. The stability of NNs with variable delays has been widely studied using the Lyapunov-Krasovskii functionals (LKFs) technique because time delays in actual NNs are usually time-varying.

    In order to analyze and solve the stability problems of DNNs, common approaches such as the Lyapunov stability method and linear matrix inequalities (LMIs) are often adopted. Among these, the Lyapunov stability method is widely applied, while LMIs provide a useful descriptive framework for these systems. This method aims to create suitable LKFs to derive less conservative stability conditions, ensuring the stability of the studied DNN even with delays varying within the largest possible closed interval. Various types of augmented LKFs were introduced in [9,10,11,12,13,14,15,16,17,18,19,20,21,22] to investigate the asymptotic or exponential stability that is dependent on delay in DNNs with varying temporal delays. Utilizing the augmented LKF approach, numerous improved stability criteria were also established. Moreover, integral inequality, the free-weighting matrix method, and reciprocally convex combination, which are commonly used methods or techniques, have been utilized to obtain the stability criteria. To reduce the conservatism of stability criteria, recent works [10,13,14,15] have employed delayed state derivative terms as augmented vector elements to estimate the time derivative. Although the existing literature indicates that the resultant stability criteria are less conservative, it is worth noting that the dimensions of the criteria in the LMI formulation experience substantial expansion. Thus, enhanced LKFs generally increase the difficulty and complexity, resulting in a corresponding increase in the server's computational burden and time.

    The work done by the above scholars still requires symmetry in the construction of LKF. In [23], two novel delay-dependent stability criteria for time-delay systems are presented, utilizing LMIs. Both are established via asymmetric augmented LKFs, ensuring positivity without the need for all involved matrices in the LKFs to be symmetric and positive definite. In [24], the author used the same method to study the stability of Takagi-Sugeno fuzzy system.

    In this paper, the primary contribution can be outlined as follows:

    (1) An improved asymmetric LKF is proposed, which can be positively definite without requiring that all matrix variables be symmetric or positive-definite.

    (2) A new stability criterion is formulated by utilizing linear matrix inequalities incorporating integral inequality and reciprocally convex combination techniques.

    (3) Compared to traditional methods, this new approach has less conservatism and complexity, which enables it to more accurately characterize neural network stability issues.

    Ultimately, the efficacy and superiority of this novel approach were successfully demonstrated, corroborating its robustness and superiority over existing methodologies through a commonly employed numerical illustration, providing a feasible solution for practical engineering applications.

    Notations: Rn denotes the n-dimensional Euclidean vector space. Rn×m is the set of all n×m real matrices. Dn+ represents the set of positive-definite diagonal matrices of Rn×n. diag{} denotes a block-diagonal matrix. He(M)=M+MT. N stands for the sets of nonnegative integers.

    Lemma 2.1. (Jensen's inequality[25]) Given Q>0, for any continuous function

    ζ(θ):[δ1,δ2]Rn,

    the following inequality holds:

    (α1α2)α1α2ζT(θ)Qζ(θ)dθ(α1α2ζ(θ)dθ)TQ(α1α2ζ(θ)dθ),
    (α1α2)22α1α2α1s˙xT(u)Q˙x(u)duds (α1α2α1s˙x(u)duds)TQ(α1α2α1s˙x(u)duds).

    Lemma 2.2. (Wirtinger-based integral inequality[26]) Given R>0, for any continuous function

    ζ(η):[δ1,δ2]Rn,

    the following inequality holds:

    δ2δ1ζT(η)(s)Rζ(η)dη1δ2δ1(δ2δ1ζT(η)dη)R(δ2δ1ζ(η)dη)+3δ2δ1ΦTRΦ,

    where

    Φ=δ2δ1ζ(η)dη2δ2δ1δ2δ1δ1θζ(η)dηdθ,δ2δ1˙ζT(η)R˙ζ(η)dηπ2(δ2δ1)2δ2δ1ζT(η)(s)Rζ(η)dη.

    Lemma 2.3. (B-L inequality[27]) Given R>0, for any continuous function

    x(t):[ψ1,ψ2]Rn,

    the following inequality holds:

    ψ2ψ1˙xT(s)R˙x(s)ds 1ψ2ψ1ϑTN[Nk=0(2k+1)ΓTN(k)RΓN(k)]ϑN

    holds for any NN, where

    ϑN:= {col{x(ψ2),x(ψ1)},                             N=0,col{x(ψ2),x(ψ1),1ψ2ψ1Ω0,,1ψ2ψ1ΩN1},N1,m=1ψ2ψ1,ΓN(k):={[I  I],                                               N=0,[I  (1)k+1I  γ0NKI  γN1NKI],              N1,γjNk:={(2j+1)(1(1)k+j),              jk1,0,                                                       jk1.

    Consider the following DNNs:

    ˙x(t)=Ax(t)+W0f(x(t))+W1f(x(th(t))), (3.1)

    where

    x(t)=[x1(t),x2(t),,xn(t)]TRn

    represents the neuron state vector, and the activation functions are given by

    f(x(t))=[f1(x1(t)),f2(x2(t)),,fn(xn(t))]T,A=diag{a1,a2,,an}>0,  Wi(i=0,1)

    are the connection matrices. ht is the abbreviation of h(t) denotes a time-varying delay, which is a differentiable function satisfying:

    0hthm, ν1˙htν2, (3.2)

    where hm, ν1 and ν2 are the real constants.

    The activation functions f() is continuous, satisfy

    fi(0)=0

    and

    k1ifi(u1)fi(u2)u1u2k2i,  u1u2, (3.3)

    where k1i and k2i are known real constants, i=1,2,,m. For convenience, we define

    K1=diag{k11,k12,,k1m}

    and

    K2=diag{k21,k22,,k2m}.

    For simplicity, we define the following notations:

    ht=h(t),  fw(t)=f(x(t)),eTi=[0n(i1)n   Inn   0n(10i)n],  i=1,2 ,10,μ1=col{x(t),x(thm)},  μ2=col{x(tht),f(x(t))},μ3=col{f(x(th(t))),tthm˙x(s)ds},μ4=col{tthmx(u)du,ts x(u)du},μ5=col{tthmtsx(u)du,tthmthmθtsx(u)dudsdθ},ζ(t)=col{μ1,μ2,μ3,μ4,μ5}.

    Theorem 3.1. Given hm>0, system (1) is asymptotically stable if there exist matrices

    Q=[Q1Q2]

    with

    Q1=QT1,   P=PT=[P11 P12 P22],P11,P12,P22Rn×n,   QiRn×n, (i=1,2),TiRn×n>0, (i=1,2),   FiRn×n>0, (i=1,2),NiDn+, (i=1,2),

    any matrices S1Rn×n,GRn×n, such that

    R7=[T1GT1]>0, (3.4)
    R8=[Q1+T112Q2T2]>0, (3.5)
    R9=[Θ11Θ1200Θ22Θ233h1mQ2Θ33Θ3412h2mT2]>0, (3.6)
    R10=[Π11Π12Π22]<0, (3.7)
    Θ11=eT1[hmP11+h2mT1]e1,Θ12=eT1[hmP12hmT1]e7,Θ22=eT7[hmP22+4Q1+4T1]e7,Θ23=eT7[2Q26h1mQ16h1mT1]e9,Θ33=eT9[12h2mQ13h1mHe(Q2)+12h2mT1+4T2]e9,Θ34=eT9[6h2mQ26h1mT2]e10,Π11=eT1[Q1+h2mT2+He(P12AP11)+h2mAT1AK1R1S1+F1+(X1)T1]e1eT2[K1R2(X1)T1+Q1]e2eT3[T2Xπ2h2mT1]e3+eT1[He(12S1GPT12)]e2+eT1[He(P22AP12+12QT2)]e3eT2[He(P22+12Q2)]e3,Π12=eT1[PT11WT0h2mAT1W0+K2R1]e4+eT1[PT11WT1h2mAT1W1]e5+eT1S1e6+eT1[GT1]e7eT2K2R2e512eT2S1e6+eT2[GTT1]e7+eT3PT12WT0e4+eT3PT12WT1e5,Π22=eT4[F2R1+h2mW0T1W0]e4+eT4[h2mHe(W0T1×W1)]e5+eT5[h2mW1T1W1R2(1˙ht)F2]e5eT6S1e6+eT7[2T1He(G)(1ht)F1]e7.

    Proof. Consider the following LKF candidate:

    Vt=4i=1Vti, (3.8)

    where

    Vt1= ρT(t)Pρ(t), ρ(t)=[xT(t)tthmxT(u)du]T,Vt2=tthmxT(s)Q[xT(s)  tsxT(u)du]Tds,  Q=[Q1 Q2],Vt3=tthtxT(s)F1x(s)ds+hmtthmts˙xT(u)T1˙x(u)duds+hmtthmtsxT(u)T2x(u)duds,Vt4=tthtfTw(s)F2fw(s)ds.

    By Lemma 2.1, since T1>0 and T2>0, we obtain

    hmts˙xT(u)T1˙x(u)du(x(t)x(s))TT1(x(t)x(s)), (3.9)
    hmtsxT(u)T2x(u)dutsxT(u)duT2tsx(u)du. (3.10)

    Thus, we can infer

    Vt2+Vt3=tthtxT(s)F1x(s)ds+tthm[xT(s)Q[xT(s)   tsxT(u)du]T+hmts[˙xT(u)T1˙x(u)+xT(u)T2x(u)]du]dstthtxT(s)F1x(s)ds+tthm[x(t)x(s)tsx(u)du]TR6  [x(t)x(s)tsx(u)du]ds, (3.11)

    where

    R6=[T1T10Q1+T112Q2T2].

    Based on the above conditions, we can get

    4i=1VtiVt1+tthm[x(t)x(s)tsx(u)du]TR6  [x(t)x(s)tsx(u)du]ds+tthtxT(s)F1x(s)ds+tthtfT(x(s))F2f(x(s))ds1hmζT(t)R9ζ(t)+tthtxT(s)F1x(s)ds+tthtfT(x(s))F2f(x(s))ds>0. (3.12)

    The time derivatives of Vti,i{1,,4} along the trajectory of system (3.1) is given by:

    ˙Vt1=2˙ρT(t)Pρ(t),˙Vt2=xT(t)Q1x(t)+tthmxT(u)duQ2x(t)xT(thm)Q[xT(thm)  tthmxT(u)du]T,˙Vt3=h2m˙xT(t)T1˙x(t)hmtthm˙xT(u)T1˙x(u)du+h2mxT(t)T2x(t)hmtthmxT(u)T2x(u)du+xT(t)F1x(t)(1˙ht)xT(tht)F1x(tht),˙Vt4=fT(x(t))F2f(x(t))(1˙ht)fT(x(tht))F2f(x(tht)). (3.13)

    For any symmetric matrices S1, the following one equalities hold:

    [x(t)x(thm)+tthm˙x(s)ds]T×S1[x(t)x(thm)tthm˙x(s)ds]=[e1e2+e6]TS1[e1e2e6]. (3.14)

    According to (3.3), for any appropriate diagonal matrices

    Ni=diag{ni1,ni2,,nim}>0,  (i=1,2),

    we have

    0ni=1[fi(xi(t))k1ixi(t)]ni1[k2ixi(t)fi(xi(t))]+ni=1[fi(xi(th(t)))k1ixi(th(t)]ni2[k2ixi(th(t))fi(xi(th(t)))]=[f(x(t))K1x(t)]TN1[K2x(t)f(x(t))]+[f(x(th(t)))K1x(th(t)]TN2[K2x(th(t))f(x(th(t)))]. (3.15)

    Then

    0[x(t)f(x(t))]T[M1N1M2N1N1][x(t)f(x(t))]+[x(th(t))f(x(th(t)))]T[M1N2M2N2N2][x(th(t))f(x(th(t)))]=[e1e4]T[M1N1M2N1N1][e1e4]+[e3e5]T[M1N2M2N2N2][e3e5], (3.16)

    where

    M1=diag{k11k21,k12k22,,k1mk2m},M2=diag{k11+k212,k12+k222,,k1m+k2m2}.

    Based on (3.4) and (3.5), we can obtain

    hmtthm˙xT(u)T1˙x(u)du=αhmtthm˙xT(u)T1˙x(u)du(1α)hmtthm˙xT(u)T1˙x(u)duα[e1e2e3]T[T1MT1+MT1T1+MT2T1MMT][e1e2e3](1α)π2hmeT7T1e7. (3.17)

    According to Lemma 2.1, we can obtain

    hmtthmxT(u)T2x(u)dutthmxT(u)duT2tthmx(u)du. (3.18)

    By combining (3.16)–(3.18), one can derive that

    ˙VtζT(t)R10ζ(t). (3.19)

    The inequality (3.19) holds, so

    ζT(t)R10ζ(t)<0

    proves that ˙Vt<0. Therefore, if LMIs (3.4)–(3.7) hold, the system (3.1) is asymptotically stable.

    The proof is completed.

    Remark 3.1. The underlying expression is deftly rescaled through the integration of the Wirtinger-based integral inequality and the B-L inequality. Here, their coefficients are assigned as α and 1α, respectively. This approach provides a flexible transformation within the range of [0,1], which facilitates the pursuit of optimal amalgamation. In scenarios where α=1, thus making 1α=0, the outcome solely relies on scaling through the B-L inequality. Conversely, if α=0, which makes 1α=1, scaling exclusively uses the Wirtinger-based integral inequality.

    Remark 3.2. Numerous researchers have introduced various LKFs in the analysis of DNNs, traditionally requiring the matrices to be symmetric. However, this study optimizes this traditional constraint by devising asymmetric forms of LKFs. With such asymmetric constructs, it is not mandatory for each matrix to be symmetric or positive definite when setting the conditions. Consequently, the conditions become more relaxed, leading to a less conservative theorem, thus broadening the horizons of analysis in this domain.

    In this part, a numerical example is given to illustrate the effectiveness of the suggested stability criterion. The primary goal is to acquire an acceptable maximum upper bound (AMUB) that is deemed acceptable for the time-varying delays and provides assurance of the neural networks under consideration's global asymptotic stability. As the AMUB increases, the stability criterion becomes less conservative.

    Example 1. We consider DNNs (3.1) with the following parameters [17,18,21,28,29]:

    W0=[0.05030.04540.09870.2075],  W1=[0.23810.93200.03880.5062]

    and

    M1=[0,0;0,0],M2=[0.3,0;0,0.8],A=[1.5,0;0,0.7].

    In Table 1, the maximum time delay achieved for various values of μ within a specific range is presented for Theorem 1 and other related articles.

    Table 1.  The maximum allowable delays of ht with various μ for Example 1.
    Methods μ=0.4 μ=0.45 μ=0.5 μ=0.55
    [17]Th.1 7.6697 6.7287 6.4126 6.2569
    [21]Th.1 8.3186 7.2119 6.8373 6.6485
    [28]Th.1 8.3958 7.3107 7.0641 6.7829
    [29]Th.1 10.1095 8.6732 8.1733 7.8993
    [18]Th.3 13.8671 11.1174 10.0050 9.4157
    Theorem 1 13.7544 12.4328 11.4954 10.7915

     | Show Table
    DownLoad: CSV

    It is evident that the AMUB as derived from Theorem 1 surpasses that obtained in the previous work by Theorem 1 presented in [17,18,21,28,29]. Compared with Theorem 3 in [26], a larger AMUB is obtained when μ ranges from 0.45 to 0.55. When μ is 0.4, the result obtained is smaller than that obtained by [26]. Importantly, the number of NVs involved in Theorem 1, as shown in Table 2, is 16n2+12n, which is less than the 79.5n2+13.5n reported in [26]. This reduction in the number of NVs lowers the computational complexity from a computation perspective. When μ = 0.4 and the initial states are 0.5, 0.8, -0.5, and -0.8, the corresponding state trajectories are illustrated in Figure 1. For μ = 0.4, the illustration shows that the trajectories tend toward zero near the abscissa of around 350. With μ = 0.55 and the same initial states, the state trajectories are depicted in Figure 2, where they tend towards zero near the abscissa of approximately 300.

    Table 2.  Computational complexity.
    Approaches Number of DVs Number of LMIs
    Th.1[17] 15n2+16n 24
    Th.1[21] 39.5n2+29.5n 40
    Th.1[28] 15n2+11n 17
    Th.1[29] 87n2+41n 46
    Th.3[18] 79.5n2+13.5n 17
    Theorem 1 16n2+12n 10

     | Show Table
    DownLoad: CSV
    Figure 1.  The state trajectories with μ=0.4.
    Figure 2.  The state trajectories with μ=0.55.

    A suitable asymmetric LKF has been constructed, enabling us to demonstrate its positive definiteness without the necessity for all matrix variables to be symmetric or positive-definite. Additionally, a novel combinatorial optimization approach is employed to its full potential, utilizing the linear combination of multiple inequalities to identify the optimal arrangement and to process these inequalities. Therefore, the conditions presented in this study are shown to have less conservatism, and our newly proposed technique exhibits tremendous potential in terms of its theoretical and empirical capability to generate larger maximum allowable delays in comparison to select recent works in the literature. Furthermore, both theoretical and quantitative analyses confirm that our method notably reduces conservatism. Finally, the proposed method can be effectively combined with the delay segmentation technique to segment the delay interval into N segments for fine processing, which will be further investigated in future work.

    Throughout the preparation of this work, we utilized the AI-based proofreading tool "Grammarly" to identify and correct grammatical errors. Subsequently, we carefully reviewed the content and made any necessary additional edits. We take complete responsibility for the content of this publication.

    This work was supported by the National Natural Science Foundation of China under Grant (No. 12061088), the Key R & D Projects of Sichuan Provincial Department of Science and Technology (2023YFG0287) and Sichuan Natural Science Youth Fund Project (Nos. 24NSFSC7038 and 2024NSFSC1404).

    There are no conflicts of interest regarding this work.

    [1] Phillips RR (2001) Crystals, defects and microstructures: Modeling across scales, Cambridge: Cambridge University Press.
    [2] Yip S (2005) Handbook of materials modeling, Berlin: Springer.
    [3] Steinhauser MO (2017) Computational Multiscale Modeling of Fluids and Solids-Theory and Applications, 2nd Edition, Berlin: Springer.
    [4] McNeil PL, Terasaki M (2001) Coping with the inevitable: how cells repair a torn surface membrane. Nat Cell Biol 3: E124–E129. doi: 10.1038/35074652
    [5] Schmidt M, Kahlert U, Wessolleck J, et al. (2014) Characterization of a setup to test the impact of high-amplitude pressure waves on living cells. Sci Rep 4: 3849.
    [6] Gambihler S, Delius M, Ellwart JW (1992) Transient increase in membrane permeability of L1210 cells upon exposure to lithotripter shock waves in vitro. Naturwissenschaften 79: 328–329. doi: 10.1007/BF01138714
    [7] Gambihler S, Delius M, Ellwart JW (1994) Permeabilization of the plasma membrane of L1210 mouse leukemia cells using lithotripter shock waves. J Membrane Biol 141: 267–275.
    [8] Kodama T, Doukas AG, Hamblin MR (2002) Shock wave-mediated molecular delivery into cells. BBA-Mol Cell Res 1542: 186–194.
    [9] Bao G, Suresh S (2003) Cell and molecular mechanics of biological materials. Nat Mater 2: 715–725. doi: 10.1038/nmat1001
    [10] Tieleman DP, Leontiadou H, Mark AE, et al. (2003) Simulation of Pore Formation in Lipid Bilayers by Mechanical Stress and Electric Fields. J Am Chem Soc 125: 6382–6383. doi: 10.1021/ja029504i
    [11] Sundaram J, Mellein BR, Mitragotri S (2003) An Experimental and Theoretical Analysis of Ultrasound-Induced Permeabilization of Cell Membranes. Biophys J 84: 3087–3101. doi: 10.1016/S0006-3495(03)70034-4
    [12] Doukas AG, Kollias N (2004) Transdermal drug delivery with a pressure wave. Adv Drug Deliver Rev 56: 559–579. doi: 10.1016/j.addr.2003.10.031
    [13] Coussios CC, Roy RA (2008) Applications of Acoustics and Cavitation to Noninvasive Therapy and Drug Delivery. Annu Rev Fluid Mech 40: 395–420. doi: 10.1146/annurev.fluid.40.111406.102116
    [14] Prausnitz MR, Langer R (2008) Transdermal drug delivery. Nat Biotechnol 26: 1261–1268. doi: 10.1038/nbt.1504
    [15] Ashley CE, Carnes EC, Phillips GK, et al. (2011) The targeted delivery of multicomponent cargos to cancer cells by nanoporous particle-supported lipid bilayers. Nat Mater 10: 389–397. doi: 10.1038/nmat2992
    [16] Koshiyama K, Wada S (2011) Molecular dynamics simulations of pore formation dynamics during the rupture process of a phospholipid bilayer caused by high-speed equibiaxial stretching. J Biomech 44: 2053–2058. doi: 10.1016/j.jbiomech.2011.05.014
    [17] Steinhauser MO (2016) On the Destruction of Cancer Cells Using Laser-Induced Shock-Waves: A Review on Experiments and Multiscale Computer Simulations. Radiol Open J 1: 60–75. doi: 10.17140/ROJ-1-110
    [18] Krehl POK (2009) History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference, Berlin: Springer.
    [19] Steinhauser MO, Schneider J, Blumen A (2009) Simulating dynamic crossover behavior of semiflexible linear polymers in solution and in the melt. J Chem Phys 130: 164902. doi: 10.1063/1.3111038
    [20] Rodriguez V, Saurel R, Jourdan G, et al. (2013) Solid-particle jet formation under shock-wave acceleration. Phys Rev E 88: 063011. doi: 10.1103/PhysRevE.88.063011
    [21] Zheng J, Chen QF, Gu YJ, et al. (2012) Hugoniot measurements of double-shocked precompressed dense xenon plasmas. Phys Rev E 86: 066406. doi: 10.1103/PhysRevE.86.066406
    [22] Falk K, Regan SP, Vorberger J, et al. (2013) Comparison between x-ray scattering and velocityinterferometry measurements from shocked liquid deuterium. Phys Rev E 87: 043112. doi: 10.1103/PhysRevE.87.043112
    [23] Brujan EA, Matsumoto Y (2014) Shock wave emission from a hemispherical cloud of bubbles in non-Newtonian fluids. J Non-Newton Fluid 204: 32–37. doi: 10.1016/j.jnnfm.2013.12.003
    [24] Iakovlev S, Iakovlev S, Buchner C, et al. (2014) Resonance-like phenomena in a submerged cylindrical shell subjected to two consecutive shock waves: The effect of the inner fluid. J Fluid Struct 50: 153–170. doi: 10.1016/j.jfluidstructs.2014.05.013
    [25] Bringa EM, Caro A,Wang YM, et al. (2005) Ultrahigh strength in nanocrystalline materials under shock loading. Science 309: 1838–1841. doi: 10.1126/science.1116723
    [26] Kadau K, Germann TC, Lomdahl PS, et al. (2007) Shock waves in polycrystalline iron. Phys Rev Lett 98: 135701. doi: 10.1103/PhysRevLett.98.135701
    [27] Knudson MD, Desjarlais MP, Dolan DH (2008) Shock-Wave Exploration of the High-Pressure Phases of Carbon. Science 322: 1822–1825. doi: 10.1126/science.1165278
    [28] Gurnett DA, Kurth WS (2005) Electron plasma oscillations upstream of the solar wind termination shock. Science 309: 2025–2027. doi: 10.1126/science.1117425
    [29] Gurnett DA, Kurth WS (2008) Intense plasma waves at and near the solar wind termination shock. Nature 454: 78–80. doi: 10.1038/nature07023
    [30] Dutton Z, Budde M, Slowe C, et al. (2001) Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein condensate. Science 293: 663–668. doi: 10.1126/science.1062527
    [31] Damski B (2006) Shock waves in a one-dimensional Bose gas: From a Bose-Einstein condensate to a Tonks gas. Phys Rev A 73: 043601. doi: 10.1103/PhysRevA.73.043601
    [32] Chang JJ, Engels P, Hoefer MA (2008) Formation of dispersive shock waves by merging and splitting Bose-Einstein condensates. Phys Rev Lett 101: 170404. doi: 10.1103/PhysRevLett.101.170404
    [33] Millot M, Dubrovinskaia N, Černok A, et al. (2015) Planetary science. Shock compression of stishovite and melting of silica at planetary interior conditions. Science 347: 418–420.
    [34] Bridge HS, Lazarus AJ, Snyder CW, et al. (1967) Mariner V: Plasma and Magnetic Fields Observed near Venus. Science 158: 1669–1673. doi: 10.1126/science.158.3809.1669
    [35] McKee CF, Draine BT (1991) Interstellar shock waves. Science 252: 397–403. doi: 10.1126/science.252.5004.397
    [36] McClure S, Dorfmüller C (2002) Extracorporeal shock wave therapy: Theory and equipment. Clin Tech Equine Pract 2: 348–357.
    [37] Lingeman JE, McAteer JA, Gnessin E, et al. (2009) Shock wave lithotripsy: advances in technology and technique. Nat Rev Urol 6: 660–670. doi: 10.1038/nrurol.2009.216
    [38] Cherenkov PA (1934) Visible emission of clean liquids by action of gamma radiation. Dokl Akad Nauk SSSR 2: 451–454.
    [39] Mach E, Salcher P (1887) Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge. Ann Phys 268: 277–291. doi: 10.1002/andp.18872681008
    [40] Kühn M, Steinhauser MO (2008) Modeling and simulation of microstructures using power diagrams: Proof of the concept. Appl Phys Lett 93: 034102. doi: 10.1063/1.2959733
    [41] Walsh JM, Rice MH (1957) Dynamic compression of liquids from measurements on strong shock waves. J Chem Phys 26: 815–823. doi: 10.1063/1.1743414
    [42] Asay JR, Chhabildas LC (2003) Paradigms and Challenges in Shock Wave Research, High-Pressure Shock Compression of Solids VI, New York: Springer-Verlag New York, 57–119.
    [43] Steinhauser MO, Grass K, Strassburger E, et al. (2009) Impact failure of granular materials-Nonequilibrium multiscale simulations and high-speed experiments. Int J Plasticity 25: 161–182. doi: 10.1016/j.ijplas.2007.11.002
    [44] Watson E, Steinhauser MO (2017) Discrete Particle Method for Simulating Hypervelocity Impact Phenomena. Materials 10: 379. doi: 10.3390/ma10040379
    [45] Holian BL, Lomdahl PS (1998) Plasticity induced by shock waves in nonequilibrium moleculardynamics simulations. Science 280: 2085–2088. doi: 10.1126/science.280.5372.2085
    [46] Kadau K, Germann TC, Lomdahl PS, et al. (2002) Microscopic view of structural phase transitions induced by shock waves. Science 296: 1681–1684. doi: 10.1126/science.1070375
    [47] Chen M, McCauley JW, Hemker KJ (2003) Shock-Induced Localized Amorphization in Boron Carbide. Science 299: 1563–1566. doi: 10.1126/science.1080819
    [48] Holian BL (2004) Molecular dynamics comes of age for shockwave research. Shock Waves 13: 489–495.
    [49] Germann TC, Kadau K (2008) Trillion-atom molecular dynamics becomes a reality. Int J Mod Phys C 19: 1315–1319. doi: 10.1142/S0129183108012911
    [50] Ciccotti G, Frenkel G, McDonald IR (1987) Simulation of Liquids and Solids, Amsterdam: North-Holland.
    [51] Allen MP, Tildesley DJ (1987) Computer Simulation of Liquids, Oxford, UK: Clarendon Press.
    [52] Liu WK, Hao S, Belytschko T, et al. (1999) Multiple scale meshfree methods for damage fracture and localization. Comp Mater Sci 16: 197–205. doi: 10.1016/S0927-0256(99)00062-2
    [53] Gates TS, Odegard GM, Frankland SJV, et al. (2005) Computational materials: Multi-scale modeling and simulation of nanostructured materials. Compos Sci Technol 65: 2416–2434. doi: 10.1016/j.compscitech.2005.06.009
    [54] Steinhauser MO (2013) Computer Simulation in Physics and Engineering, 1st Edition, Berlin: deGruyter.
    [55] Finnis MW, Sinclair JE (1984) A simple empirical N-body potential for transition metals. Philos Mag A 50: 45–55. doi: 10.1080/01418618408244210
    [56] Kohn W (1996) Density functional and density matrix method scaling linearly with the number of atoms. Phys Rev Lett 76: 3168–3171. doi: 10.1103/PhysRevLett.76.3168
    [57] Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Lett 55: 2471–2474. doi: 10.1103/PhysRevLett.55.2471
    [58] Elstner M, Porezag D, Jungnickel G, et al. (1998) Self-consistent-charge density-functional tightbinding method for simulations of complex materials properties. Phys Rev B 58: 7260–7268. doi: 10.1103/PhysRevB.58.7260
    [59] Sutton AP, Finnis MW, Pettifor DG, et al. (1988) The tight-binding bond model. J Phys C-Solid State Phys 21: 35–66. doi: 10.1088/0022-3719/21/1/007
    [60] Szabo A, Ostlund NS (1996) Modern quantum chemistry: introduction to advanced electronic structure theory, (Dover Books on Chemistry), New York: Dover Publications.
    [61] Kadau K, Germann TC, Lomdahl PS (2006) Molecular dynamics comes of age: 320 billion atom simulation on BlueGene/L. Int J Mod Phys C 17: 1755–1761. doi: 10.1142/S0129183106010182
    [62] Fineberg J (2003) Materials science: close-up on cracks. Nature 426: 131–132. doi: 10.1038/426131a
    [63] Buehler M, Hartmaier A, Gao H, et al. (2004) Atomic plasticity: description and analysis of a onebillion atom simulation of ductile materials failure. Comput Method Appl M 193: 5257–5282. doi: 10.1016/j.cma.2003.12.066
    [64] Abraham FF, Gao HJ (2000) How fast can cracks propagate? Phys Rev Lett 84: 3113–3116. doi: 10.1103/PhysRevLett.84.3113
    [65] Bulatov V, Abraham FF, Kubin L, et al. (1998) Connecting atomistic and mesoscale simulations of crystal plasticity. Nature 391: 669–672. doi: 10.1038/35577
    [66] Gross SP, Fineberg J, Marder M, et al. (1993) Acoustic emissions from rapidly moving cracks. Phys Rev Lett 71: 3162–3165. doi: 10.1103/PhysRevLett.71.3162
    [67] Courant R (1943) Variational Methods for the Solution of Problems of Equilibrium and Vibrations. B Am Math Soc 49: 1–23.
    [68] Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82: 1013–1024. doi: 10.1086/112164
    [69] Cabibbo N, Iwasaki Y, Schilling K (1999) High performance computing in lattice QCD. Parallel Comput 25: 1197–1198. doi: 10.1016/S0167-8191(99)00045-9
    [70] Evertz HG (2003) The loop algorithm. Adv Phys 52: 1–66. doi: 10.1080/0001873021000049195
    [71] Holm EA, Battaile CC (2001) The computer simulation of microstructural evolution. JOM 53: 20–23.
    [72] Nielsen SO, Lopez CF, Srinivas G, et al. (2004) Coarse grain models and the computer simulation of soft materials. J Phys-Condens Mat 16: 481–512. doi: 10.1088/0953-8984/16/15/R03
    [73] Praprotnik M, Site LD, Kremer K (2008) Multiscale simulation of soft matter: From scale bridging to adaptive resolution. Annu Rev Phys Chem 59: 545–571. doi: 10.1146/annurev.physchem.59.032607.093707
    [74] Karimi-Varzaneh HA, Müller-Plathe F (2011) Coarse-Grained Modeling for Macromolecular Chemistry, In: Kirchner B, Vrabec J, Topics in Current Chemistry, Berlin, Heidelberg: Springer, 326–321.
    [75] Müller-Plathe F (2002) Coarse-graining in polymer simulation: from the atomistic to the mesoscopic scale and back. Chem Phys Chem 3: 755–769.
    [76] Abraham FF, Broughton JQ, Broughton JQ, et al. (1998) Spanning the length scales in dynamic simulation. Comp Phys 12: 538–546. doi: 10.1063/1.168756
    [77] Abraham FF, Brodbeck D, Rafey R, et al. (1994) Instability dynamics of fracture: A computer simulation investigation. Phys Rev Lett 73: 272–275. doi: 10.1103/PhysRevLett.73.272
    [78] Abraham FF, Brodbeck D, Rudge WE, et al. (1998) Ab initio dynamics of rapid fracture. Model Simul Mater Sc 6: 639–670. doi: 10.1088/0965-0393/6/5/010
    [79] Warshel A, LevittM(1976) Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. J Mol Biol 103: 227–249.
    [80] Winkler RG, Steinhauser MO, Reineker P (2002) Complex formation in systems of oppositely charged polyelectrolytes: a molecular dynamics simulation study. Phys Rev E 66: 021802.
    [81] Dünweg B, Reith D, Steinhauser M, et al. (2002) Corrections to scaling in the hydrodynamic properties of dilute polymer solutions. J Chem Phys 117: 914–924. doi: 10.1063/1.1483296
    [82] Stevens MJ (2004) Coarse-grained simulations of lipid bilayers. J Chem Phys 121: 11942–11948. doi: 10.1063/1.1814058
    [83] Steinhauser MO (2005) A molecular dynamics study on universal properties of polymer chains in different solvent qualities. Part I. A review of linear chain properties. J Chem Phys 122: 094901.
    [84] Steinhauser MO, Hiermaier S (2009) A Review of Computational Methods in Materials Science: Examples from Shock-Wave and Polymer Physics. Int J Mol Sci 10: 5135–5216. doi: 10.3390/ijms10125135
    [85] Goetz R, Gompper G, Lipowsky R (1999) Mobility and elasticity of self-assembled membranes. Phys Rev Lett 82: 221–224. doi: 10.1103/PhysRevLett.82.221
    [86] Lipowsky R (2004) Biomimetic membrane modelling: pictures from the twilight zone. Nat Mater 3: 589–591. doi: 10.1038/nmat1208
    [87] Lyubartsev AP (2005) Multiscale modeling of lipids and lipid bilayers. Eur Biophys J 35: 53–61. doi: 10.1007/s00249-005-0005-y
    [88] Orsi M, Michel J, Essex JW (2010) Coarse-grain modelling of DMPC and DOPC lipid bilayers. J Phys-Condens Mat 22: 155106. doi: 10.1088/0953-8984/22/15/155106
    [89] Steinhauser MO (2012) Introduction to Molecular Dynamics Simulations: Applications in Hard and Soft Condensed Matter Physics, InTech.
    [90] Alberts B, Bray D, Johnson A, et al. (2000) Molecular Biology of the Cell, 4 Edition, New York: Garland Science, Taylor and Francis Group.
    [91] Steinhauser MO, Steinhauser MO, Schmidt M (2014) Destruction of cancer cells by laserinduced shock waves: recent developments in experimental treatments and multiscale computer simulations. Soft Matter 10: 4778–4788. doi: 10.1039/C4SM00407H
    [92] Tozzini V (2004) Coarse-grained models for proteins. Curr Opin Struc Biol 15: 144–150.
    [93] Ayton GS, Noid WG, Voth GA (2007) Multiscale modeling of biomolecular systems: in serial and in parallel. Curr Opin Struc Biol 17: 192–198. doi: 10.1016/j.sbi.2007.03.004
    [94] Forrest LR, Sansom MS (2000) Membrane simulations: bigger and better? Curr Opin Struc Biol 10: 174–181. doi: 10.1016/S0959-440X(00)00066-X
    [95] Woods CJ, Mulholland AJ (2008) Multiscale modelling of biological systems. RSC Special Periodicals Report: Chemical Modelling, Applications and Theory 5: 13–50. doi: 10.1039/b608778g
    [96] Steinhauser MO (editor) (2016) Special Issue of the Journal Materials: Computational Multiscale Modeling and Simulation in Materials Science. Available from: http://www.mdpi.com/journal/materials/special issues/modeling and simulation.
    [97] Brendel W (1986) Shock Waves: A New Physical Principle in Medicine. Eur Surg Res 18: 177–180. doi: 10.1159/000128523
    [98] Wang CJ (2003) An overview of shock wave therapy in musculoskeletal disorders. Chang Gung Med J 26: 220–232.
    [99] Wang ZJZ, DesernoM (2010) A systematically coarse-grained solvent-free model for quantitative phospholipid bilayer simulations. J Phys Chem B 114: 11207–11220. doi: 10.1021/jp102543j
    [100] Wang ZB, Wu J, Fang LQ, et al. (2011) Preliminary ex vivo feasibility study on targeted cell surgery by high intensity focused ultrasound (HIFU). Ultrasonics 51: 369–375. doi: 10.1016/j.ultras.2010.11.002
    [101] Wang S, Frenkel V, Zderic V (2011) Optimization of pulsed focused ultrasound exposures for hyperthermia applications. J Acoust Soc Am 130: 599–609. doi: 10.1121/1.3598464
    [102] Paul W, Smith GD, Yoon DY (1997) Static and dynamic properties of an-C100H202 melt from molecular dynamics simulations. Macromolecules 30: 7772–7780. doi: 10.1021/ma971184d
    [103] Kreer T, Baschnagel J, Mueller M, et al. (2001) Monte Carlo Simulation of long chain polymer melts: Crossover from Rouse to reptation dynamics. Macromolecules 34: 1105–1117. doi: 10.1021/ma001500f
    [104] Krushev S, Paul W, Smith GD (2002) The role of internal rotational barriers in polymer melt chain dynamics. Macromolecules 35: 4198–4203. doi: 10.1021/ma0115794
    [105] Bulacu M, van der Giessen E (2005) Effect of bending and torsion rigidity on self-diffusion in polymer melts: A molecular-dynamics study. J Chem Phys 123: 114901. doi: 10.1063/1.2035086
    [106] Kratky O, Porod G (1949) Röntgenuntersuchung gelöster Fadenmoleküle. Recl Trva Chim Pays-Bas 68: 1106–1122.
    [107] Doi M, Edwards SF (1986) The Theory of Polymer Dynamics, Oxford: Clarendon Press.
    [108] Harris RA, Hearst JE (1966) On Polymer Dynamics. J Chem Phys 44: 2595–2602. doi: 10.1063/1.1727098
    [109] Hearst JE, Harris RA (1967) On Polymer Dynamics. III. Elastic Light Scattering. J Chem Phys 46: 398–398.
    [110] Harnau L, Winkler RG, Reineker P (1997) Influence of stiffness on the dynamics of macromolecules in a melt. J Chem Phys 106: 2469–2476. doi: 10.1063/1.473154
    [111] Harnau L, WInkler RG, Reineker P (1999) On the dynamics of polymer melts: Contribution of Rouse and bending modes. EPL 45: 488–494. doi: 10.1209/epl/i1999-00193-6
    [112] Steinhauser MO (2008) Static and dynamic scaling of semiflexible polymer chains-a molecular dynamics simulation study of single chains and melts. Mech Time-Depend Mat 12: 291–312. doi: 10.1007/s11043-008-9062-9
    [113] Guenza M (2003) Cooperative dynamics in semiflexibile unentangled polymer fluids. J Chem Phys 119: 7568–7578. doi: 10.1063/1.1606674
    [114] Piran T (2004) Statistical Mechanics of Membranes and Interfaces, 2 edition, World Scientific Publishing Co., Inc.
    [115] Schindler T, Kröner D, Steinhauser MO (2016) On the dynamics of molecular self-assembly and the structural analysis of bilayer membranes using coarse-grained molecular dynamics simulations. BBA-Biomembranes 1858: 1955–1963. doi: 10.1016/j.bbamem.2016.05.014
    [116] Brannigan G, Lin LCL, Brown FLH (2006) Implicit solvent simulation models for biomembranes. Eur Biophys J 35: 104–124. doi: 10.1007/s00249-005-0013-y
    [117] Chang R, Ayton GS, Voth GA (2005) Multiscale coupling of mesoscopic- and atomistic-level lipid bilayer simulations. J Chem Phys 122: 244716. doi: 10.1063/1.1931651
    [118] Huang MJ, Kapral R, Mikhailov AS, et al. (2012) Coarse-grain model for lipid bilayer selfassembly and dynamics: Multiparticle collision description of the solvent. J Chem Phys 137: 055101. doi: 10.1063/1.4736414
    [119] Pandit SA, Scott HL (2009) Multiscale simulations of heterogeneous model membranes. BBA-Biomembranes 1788: 136–148. doi: 10.1016/j.bbamem.2008.09.004
    [120] Farago O (2003) "Water-free" computer model for fluid bilayer membranes. J Chem Phys 119: 596–605. doi: 10.1063/1.1578612
    [121] Brannigan G, Philips PF, Brown FLH (2005) Flexible lipid bilayers in implicit solvent. Phys Rev E 72: 011915. doi: 10.1103/PhysRevE.72.011915
    [122] Yuan H, Huang C, Li J, et al. (2010) One-particle-thick, solvent-free, coarse-grained model for biological and biomimetic fluid membranes. Phys Rev E 82: 011905. doi: 10.1103/PhysRevE.82.011905
    [123] Noguchi H (2011) Solvent-free coarse-grained lipid model for large-scale simulations. J Chem Phys 134: 055101. doi: 10.1063/1.3541246
    [124] Weiner SJ, Kollman PA, Case DA, et al. (1984) A new force field for molecular mechanical simulation of nucleic acids and proteins. J Am Chem Soc 106: 765–784. doi: 10.1021/ja00315a051
    [125] Paul W, Yoon DY, Smith GD, et al. (1995) An Optimized United Atom Model for Simulations of Polymethylene Melts. J Chem Phys 103: 1702–1709. doi: 10.1063/1.469740
    [126] Siu SWI, Vácha R, Jungwirth P, et al. (2008) Biomolecular simulations of membranes: physical properties from different force fields. J Phys Chem 128: 125103. doi: 10.1063/1.2897760
    [127] Drouffe JM, Maggs AC, Leibler S, et al. (1991) Computer simulations of self-assembled membranes. Science 254: 1353–1356. doi: 10.1126/science.1962193
    [128] Goetz R, Lipowsky R (1998) Computer simulations of bilayer membranes: Self-assembly and interfacial tension. J Chem Phys 108: 7397–7409. doi: 10.1063/1.476160
    [129] Noguchi H, Takasu M (2001) Self-assembly of amphiphiles into vesicles: A Brownian dynamics simulation. Phys Rev E 64: 041913. doi: 10.1103/PhysRevE.64.041913
    [130] Bourov GK, Bhattacharya A (2005) Brownian dynamics simulation study of self-assembly of amphiphiles with large hydrophilic heads. J Chem Phys 122: 44702. doi: 10.1063/1.1834495
    [131] Steinhauser MO, Grass K, Thoma K, et al. (2006) Impact dynamics and failure of brittle solid states by means of nonequilibrium molecular dynamics simulations. EPL 73: 62–68. doi: 10.1209/epl/i2005-10353-2
    [132] Yang S, Qu J (2014) Coarse-grained molecular dynamics simulations of the tensile behavior of a thermosetting polymer. Phys Rev E 90: 012601. doi: 10.1103/PhysRevE.90.012601
    [133] Eslami H, Müller-Plathe F (2013) How thick is the interphase in an ultrathin polymer film? Coarse-grained molecular dynamics simulations of polyamide-6,6 on graphene. J Phys Chem 117: 5249–5257.
    [134] Ganzenm¨uller GC, Hiermaier S, Steinhauser MO (2011) Shock-wave induced damage in lipid bilayers: a dissipative particle dynamics simulation study. Soft Matter 7: 4307–4317. doi: 10.1039/c0sm01296c
    [135] Huang WX, Chang CB, Sung HJ (2012) Three-dimensional simulation of elastic capsules in shear flow by the penalty immersed boundary method. J Comput Phys 231: 3340–3364. doi: 10.1016/j.jcp.2012.01.006
    [136] Pazzona FG, Demontis P (2012) A grand-canonical Monte Carlo study of the adsorption properties of argon confined in ZIF-8: local thermodynamic modeling. J Phys Chem 117: 349–357.
    [137] Pogodin S, Baulin VA (2010) Coarse-grained models of phospholipid membranes within the single chain mean field theory. Soft Matter 6: 2216–2226. doi: 10.1039/b927437e
    [138] Wang Y, Sigurdsson JK, Brandt E, et al. (2013) Dynamic implicit-solvent coarse-grained models of lipid bilayer membranes: fluctuating hydrodynamics thermostat. Phys Rev E 88: 023301. doi: 10.1103/PhysRevE.88.023301
    [139] Koshiyama K, Kodama T, Yano T, et al. (2006) Structural Change in Lipid Bilayers and Water Penetration Induced by Shock Waves: Molecular Dynamics Simulations. Biophys J 91: 2198–2205. doi: 10.1529/biophysj.105.077677
    [140] Koshiyama K, Kodama T, Yano T, et al. (2008) Molecular dynamics simulation of structural changes of lipid bilayers induced by shock waves: Effects of incident angles. BBA-Biomembranes 1778: 1423–1428. doi: 10.1016/j.bbamem.2008.03.010
    [141] Lechuga J, Drikakis D, Pal S (2008) Molecular dynamics study of the interaction of a shock wave with a biological membrane. Int J Numer Mech Fluids 57: 677–692. doi: 10.1002/fld.1588
    [142] Kodama T, Kodama T, Hamblin MR, et al. (2000) Cytoplasmic molecular delivery with shock waves: importance of impulse. Biophys J 79: 1821–1832. doi: 10.1016/S0006-3495(00)76432-0
    [143] Doukas AG, McAuliffe DJ, Lee S, et al. (1995) Physical factors involved in stress-wave-induced cell injury: The effect of stress gradient. Ultrasound Med Biol 21: 961–967. doi: 10.1016/0301-5629(95)00027-O
    [144] Doukas AG, Flotte TJ (1996) Physical characteristics and biological effects of laser-induced stress waves. Ultrasound Med Biol 22: 151–164. doi: 10.1016/0301-5629(95)02026-8
    [145] Lee S, Doukas AG (1999) Laser-generated stress waves and their effects on the cell membrane. IEEE J Sel Top Quant 5: 997–1003. doi: 10.1109/2944.796322
    [146] Español P (1997) Dissipative Particle Dynamics with energy conservation. EPL 40: 631–636. doi: 10.1209/epl/i1997-00515-8
    [147] Steinhauser MO, Schindler T (2017) Particle-based simulations of bilayer membranes: selfassembly, structural analysis, and shock-wave damage. Comp Part Mech 4: 69–86. doi: 10.1007/s40571-016-0126-3
    [148] Hansen JP, McDonald IR (2005) Theory of Simple Liquids, Academic Press.
  • This article has been cited by:

    1. Weizhong Chen, Haoxu Wang, Xiaohua Wang, Lei Zhang, Interval Estimation for Switched Neural Networks by Zonotopes, 2025, 0890-6327, 10.1002/acs.4042
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(8324) PDF downloads(1087) Cited by(3)

Figures and Tables

Figures(22)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog