Robert Lipton

Louisiana State University | LSU · Department of Mathematics

The hybrid particle-based computational platform that couples peridynamics with the discrete element method (PeriDEM) is used to model vehicle mobility over roadbeds. We consider wheels rolling over gravel beds, where gravel is allowed to deform and fracture. The motion of particles are not constrained to translation and rotation as in DEM and grai...

The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving f...

We introduce a nonlocal model of peridynamic type for fracture evolution in the quasistatic regime. Nonlocal quasistatic fracture evolution is developed and supporting numerical examples are presented. The approach is implicit and is based on local stationary and fixed point methods. Here a smooth cohesive force-strain model is used. Initially the...

Partition of unity methods are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a partition of unity method scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic model is used in region...

Nonlocal quasistatic fracture evolution for interacting cracks is developed and supporting numerical examples are presented. The approach is implicit and is based on local stationarity and fixed point methods. It is proved that the fracture evolution decreases stored elastic energy with each load step as the cracks advance; provided the load increm...

The Generalized Finite Element Methods (GFEM) are known for accurately resolving local features in heterogeneous media. Recent numerical experiments have shown that the use of local bases made from A-harmonic extensions of well chosen boundary data have nearly optimal approximation properties. In this paper we show that the explanation is geometric...

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discreti...

A study of the evolution of wave dispersion in systems of all‐metallic periodic structures with increasing corrugation depth shows a similarity of the properties of waves in metamaterial slow wave structures (MSWSs) and traditional metallic slow wave structures (SWSs) used in high‐power microwave (HPM) sources. This chapter aims to identify MSWSs w...

In this chapter, the authors present a perturbation analysis of Maxwell's equations to describe the interaction of an electron beam with a metamaterial a slow wave structure. They apply a two‐scale perturbation analysis to the Maxwell system used to model beam wave interaction inside the infinitely long amplifier. The authors model a finite‐size tr...

We apply a nonlinear-nonlocal field theory for numerical calculation of quasistatic fracture. The model is given by a regularized nonlinear pairwise (RNP) potential in a peridynamic formulation. The potential function is given by an explicit formula with and explicit first and second derivatives. This fact allows us to write the entries of the tang...

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM scheme for handling problems with zones of linear elasticity and zones where fractures occur. Here, the peridynamic (PD) model is used in regions of fractu...

Usage, manipulation, transport, delivery, and mixing of granular or particulate media, comprised of spherical or polyhedral particles, is commonly encountered in industrial sectors of construction (cement and rock fragments), pharmaceutics (tablets), and transportation (ballast). Elucidating particulate media’s behavior in concert with particle att...

A nonlocal field theory of peridynamic type is applied to model the brittle fracture problem. The elastic fields obtained from the nonlocal model are shown to converge in the limit of vanishing non-locality to solutions of classic plane elastodynamics associated with a running crack. We carry out our analysis for a plate subject to mode one loading...

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discreti...

On modern supercomputers, asynchronous many task systems are emerging to address the new architecture of computational nodes. Through this shift of increasing cores per node, a new programming model with focus on handling of the fine-grain parallelism with increasing amount of cores per computational node is needed. Asynchronous Many Task (AMT) run...

A simple nonlocal field theory of peridynamic type is applied to model brittle fracture. The kinetic relation for the crack tip velocity given by Linear Elastic Fracture Mechanics (LEFM) is recovered directly from the nonlocal dynamics, this is seen both theoretically and in simulations. An explicit formula for the change of internal energy inside...

Usage, manipulation, transport, delivery, and mixing of granular or particulate media, comprised of spherical or polyhedral particles, is commonly encountered in industrial sectors of construction (cement and rock fragments), pharmaceutics (tablets), and transportation (ballast). Elucidating the behavior of particulate media, in concert with partic...

We explain the sharp Lorentz resonances in plasmonic crystals that consist of 2D nano dielectric inclusions as the interaction between resonant material properties and geometric resonances of electrostatic nature. One example of such plasmonic crystals are graphene nanosheets that are periodically arranged within a non-magnetic bulk dielectric. We...

In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in Babuška and Lipton (2011). We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cos...

A nonlocal field theory is applied to model brittle fracture. The fracture evolution is shown to converge in the limit of vanishing nonlocality to classic plane elastodynamics with a running crack. We carry out our analysis for a single crack in a plate subject to mode one loading.

We identify explicit conditions on geometry and material contrast for creating band gaps in 2D photonic and 3D acoustic crystals made from composites. This approach is new and makes use of the electrostatic and quasiperiodic source-free resonances of the crystal. The source-free modes deliver a spectral representation for solution operators associa...

In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babu\v{s}ka and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373--406]. We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM so...

A mechanical model is introduced for predicting the initiation and evolution of complex fracture patterns without the need for a damage variable or law. The model, a continuum variant of Newton’s second law, uses integral rather than partial differential operators where the region of integration is over finite domain. The force interaction is deriv...

We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions b...

A simple nonlocal field theory of peridynamic type is applied to model brittle fracture. The fracture evolution is shown to converge in the limit of vanishing nonlocality to classic plane elastodynamics with a running crack. The kinetic relation for the crack is recovered directly from the nonlocal model in the limit of vanishing nonlocality. We ca...

We establish the a-priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of nonlocal interaction. The pairwise interactions betwe...

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a state-based peridynamic model using two potentials: one associated with hydrostatic strain and the other a...

We introduce a simple model for free damage propagation based on non-local potentials. The model is developed using a state based peridynamic formulation. The resulting evolution is shown to be well posed. At each instant of the evolution we identify the damage set. On this set the local strain has exceeded critical values either for tensile or hyd...

We construct metamaterials from sub-wavelength nonmagnetic resonators and consider the refraction of incoming signals traveling from free space into the metamaterial. We show that the direction of the transmitted signal is a function of its center frequency and bandwidth. The directionality of the transmitted signal and its frequency dependence is...

We analyze the time harmonic Maxwell's equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size η of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeabil...

Peridynamics is a non-local generalization of continuum mechanics tailored to address discontinuous displacement fields arising in fracture mechanics. As many non-local approaches, peridynamics requires considerable computing resources to solve practical problems. Several implementations of peridynamics utilizing CUDA, OpenCL, and MPI were develope...

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a state-based peridynamic model using two potentials: one associated with hydrostatic strain and the other...

In this work, we calculate the convergence rate of the finite difference approximation for a class of nonlocal models of peridynamic type. We consider a nonlinear bond softening model characterized by a double well potential. We show the existence of a solution in H\"{o}lder space with H\"{o}lder exponent $\gamma \in (0,1]$. The rate of convergence...

In this work, we calculate the convergence rate of the finite difference approximation for a class of nonlocal fracture models. We consider two point force interactions characterized by a double well potential. We show the existence of a evolving displacement field in Hölder space with Hölder exponent γ ∈ (0, 1]. The rate of convergence of the fini...

A model for dynamic damage propagation is developed using nonlocal potentials. The model is posed using a state-based peridynamic formulation. The resulting evolution is seen to be well posed. At each instant of the evolution, we identify a damage set. On this set, the local strain has exceeded critical values either for tensile or hydrostatic stra...

In this chapter we present a rigorous convergence analysis of finite difference and finite element approximation of nonlinear nonlocal models. In the previous chapter, we considered a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175–209, 2000). There we showed, for a fixed horizon of nonloc...

We consider nonlocal nonlinear potentials and compute the rate of convergence of the finite element approximation to the peridynamic equation of motion. The present work is a continuation and extension of the work where the finite difference approximation of peridynamics equation was shown to converge at the rate $O(h^\gamma/\epsilon)$ in H\"older...

We investigate the error incurred in replacing a nonlinear nonlocal bond based peridynamic model with linearized peridynamics or classic local elastodynamics away from the fracture set. The nonlinear nonlocal model is characterized by a double well potential. We establish a convergence rate for differentiable solutions of nonlinear nonlocal peridyn...

Motivated by the numerical experiments carried out in [S. C. Yurt, A. Elfrgani, M. I. Fuks, K. Ilyenko, and E. Schamiloglu, IEEE Trans. Plasma Sci., 44(2016), pp. 1280-1286], we apply an asymptotic analysis to show that corrugated waveguides can be approximated by smooth cylindrical waveguides with an effective metamaterial surface impedance. We sh...

We take a mesoscopic approach to dynamic fracture and formulate a nonlocal cohesive model for assessing the deformation state inside a cracking body. In this model a more complete set of physical properties including elastic and softening behavior are assigned to each point in the medium. We we work within the peridynamic framework where strains ar...

We identify explicit conditions on geometry and material contrast for creating band gaps in 2-d photonic and 3-d acoustic crystals. This approach is novel and makes use of the electro-static and quasi-periodic source free resonances of the crystal. The source free modes deliver a spectral representation for solution operators associated with propag...

We introduce a systematic method for identifying the worst case load among all boundary loads of fixed energy. Here the worst case load is defined to be the one that delivers the largest fraction of input energy to a prescribed subdomain of interest. The worst case load is identified with the first eigenfunction of a suitably defined eigenvalue pro...

We introduce a systematic method for identifying the worst case load among all boundary loads of fixed energy. Here the worst case load delivers the largest fraction of input energy to a prescribed subdomain of interest. The worst case load is identified with the first eigenfunction of a suitably defined eigenvalue problem. The first eigenvalue for...

A mechanical model is introduced for predicting the initiation and evolution of complex fracture patterns without the need for a damage variable or law. The model, a continuum variant of Newton's second law, uses integral rather than partial differential operators where the region of integration is over finite domain. The force interaction is deriv...

Analytic representation formulas and power series are developed describing the band structure inside periodic photonic and acoustic crystals made from high contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasi-periodic source free modes. These modes are used to represent solution opera...

The method of two scale convergence is implemented to study the homogenization of time-dependent nonlocal continuum models of heterogeneous media. Two integro-differential models are considered: the nonlocal convection-diffusion equation and the state-based peridynamic model in nonlocal continuum mechanics. The asymptotic analysis delivers both hom...

We employ metamaterial beam-wave interaction structures for tuning the gain and bandwidth of short traveling wave tubes. The interaction structures are made from metal rings of uniform cross section, which are periodically deployed along the length of the traveling wave tube. The aspect ratio of the ring cross sections is adjusted to control both g...

A multiscale spectral generalized finite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method...

The approach taken here solves the Maxwell equations inside metamaterial crystals directly and explicitly with no approximations made. The Bloch wave solution and dispersion relation is given by a power series in the ratio between wave number and period. Each term is iteratively defined by the solution of an auxiliary problem depending on the confi...

The Publisher regrets that this article is an accidental duplication of an article that has already been published, http://dx.doi.org/10.1016/j.photonics.2013.07.005. The duplicate article has therefore been withdrawn.

We consider the nonlocal formulation of continuum mechanics described by peridynamics. We provide a link between peridynamic evolution and brittle fracture evolution for a broad class of peridynamic potentials associated with unstable peridynamic constitutive laws. Distinguished limits of peridynamic evolutions are identified that correspond to van...

A methodology is presented for bounding the higher LpLp norms, 2⩽p⩽∞2⩽p⩽∞, of the local strain inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of...

We present a new multi-scale model for linking higher order microstructure descriptions to failure initiation and damage propagation in polycrystalline media. The model gives an accurate local field description for predicting damage nucleation at the length scale of the polycrystalline texture. The new method allows the recovery of the local damage...

A metamaterial with frequency dependent double negative effective properties is constructed from a sub-wavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Dire...

A generic class of metamaterials is introduced and is shown to exhibit frequency dependent double negative effective properties. We develop a rigorous method for calculating the frequency intervals where either double negative or double positive effective properties appear and show how these intervals imply the existence of propagating Bloch waves...

A new variational methodology is developed for computing optimal bounds on the stress inside thermoelastic composites. The method also provides tight bounds on the strength domains for random two phase elastic plastic composites. A second effort develops a global local infinite element method for problems with multiple length scales such as functio...

Multi-phase particle reinforced heat conducting composites are considered. We treat the case when there is an interfacial thermal barrier between phases. We indicate how the particle size distribution and particle geometry influences the overall properties of the suspension. These results apply immediately to the mathematically analogous case of DC...

We provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power-law behavior. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplifica...

We examine the composition of the $L^{\infty}$ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscil...

A system of periodic poly-disperse coated nano-rods is considered. Both the coated nano-rods and host material are non-magnetic. The exterior nano-coating has a frequency dependent dielectric constant and the rod has a high dielectric constant. A negative effective magnetic permeability is generated near the Mie resonances of the rods while the coa...

We obtain convergent power series representations for Bloch waves in periodic high-contrast media. The material coefficient in the inclusions can be positive or negative. The small expansion parameter is the ratio of period cell width to wavelength, and the coefficient functions are solutions of the cell problems arising from formal asymptotic expa...

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodol...

The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in $L^\infty(\Omega)$. This class of coefficients includes as examples media with micro-structure as we...

The effective properties of a two-phase composite depends both on the constituent materials and the microstructure. Starting from David J. Bergman's work on the integral representation formula (IRF) approach for deriving bounds on dielectric properties of composites with isotropic constituents, followed by Golden & Papanicolaou's work on establishi...

We obtain a convergent power series expansion for the first branch of the dispersion relation for subwavelength plasmonic crystals consisting of plasmonic rods with frequency-dependent dielectric permittivity embedded in a host medium with unit permittivity. The expansion parameter is $\eta=kd=2\pi d/\lambda$, where $k$ is the norm of a fixed wavev...

Mechanics of the composite repair under tensile loading with and without overlay plies was examined for nontraditional patch ply orientations. Three-dimensional nonlinear analysis was performed for repair failure prediction and good baseline comparison for open hole scarfed panels and panels repaired by using standard ply-by-ply replacement patch c...

A methodology is presented for bounding all higher moments of the local hydrostatic stress field inside random two phase linear thermoelastic media undergoing macroscopic thermomechanical loading. The method also provides a lower bound on the maximum local stress. Explicit formulas for the optimal lower bounds are found that are expressed in terms...

We provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power law behavior. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplifica...

A methodology is presented for bounding the higher L p norms, 2≤p≤∞, of the local stress inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the l...

We present a new method for the design of graded composite structures for strength and stiffness. The method proceeds in two steps. In the first step a new type of relaxed problem is formulated involving effective elastic tensors as well as a new multiscale quantity dubbed the macrostress modulation function. The modulation function quantifies the...

The penetration function measures the effect of the boundary data on the energy of the solution of a second order linear elliptic PDE taken over an interior subdomain. Here the coefficients of the PDE are functions of position and often represent the material properties of non homogeneous media with microstructure. The penetration function is used...

We introduce a new mathematically rigorous high fidelity asymptotic theory for recovering the local field behavior inside complex composite architectures. The theory applies to zones containing strong spatial variance of local material properties. The method is used to recover the local field across ply interfaces for a pre-stressed multi-ply fiber...

In this paper we derive bounds for the torsional rigidity of a cylindrical shaft with arbitrary transverse cross-section containing a number of cylindrically orthotropic fibres or coated fibres. The exact upper and lower bounds depend on the constituent shear rigidities, the area fractions, the locations of the reinforcements as well as the geometr...

We introduce asymptotic expansions for recovering the local field behavior inside multiscale composite architectures in the presence of residual stress. The theory applies to zones containing abrupt changes in the composite microgeometry. This includes the interfaces between plies inside fiber reinforced laminates. The asymptotic expansions are use...

We introduce a rigorously based numerical method for compliance minimization problems in the presence of pointwise stress constraints. The method is based on new multiscale quantities that measure the amplification of the local stress due to the microstructure. The design method is illustrated for two different kinds of problems. The first identifi...

The completed effort integrated the rigorous microlevel (fiber, matrix, sizing) stress bound recently obtained by Lipton [1,2] under AFOSR sponsorship with the AFRL-developed ply level multibasis spline approximation stress analysis tools [3-5]. A robust multiscale analysis framework was developed and applied within and beyond the scope of the pres...

In this paper, we model and compute flow-induced mechanical properties of nematic polymer nano-composites, consisting of transversely isotropic rigid spheroids in an isotropic matrix. Our goal is to fill a gap in the theoretical literature between random and perfectly aligned spheroidal composites (Odegard etal. in Compos. Sci. Technol. 63, 1671–16...

An extension of current methodologies is introduced for optimization of graded microstructure subject to local stress criteria. The method is based on new multiscale stress criteria given by macrostress modulation functions. The modulation functions quantify the intensity of local stress fluctuations at the scale of the microstructure due to the im...

Composites made from two linear isotropic elastic materials are considered. It is assumed that only the volume fraction of each elastic material is known. The composite is subjected to a uniform hydrostatic strain. For this case lower bounds on all rth moments of the dilatational strain field inside each phase are obtained for r⩾2. A lower bound on...

A multi-scale characterization of the field concentrations inside composite and polycrystalline media is developed. The analysis focuses on gradient fields associated with the intensive quantities given by the temperature and the electric potential. In the linear regime these quantities are modeled by the solution of a second order elliptic partial...

A new inverse homogenization procedure is applied to design graded fiber reinforced shafts subject to local stress criteria. The method is based on new multiscale stress criteria given by macrostress modulation functions. The modulation functions quantify the intensity of local stress fluctuations at the scale of the microstructure due to the impos...

Nematic, or liquid-crystalline, polymer nanocomposites (NPNCs) are composed of large aspect ratio, rod-like or platelet, rigid macromolecules in a matrix or solvent, which itself may be aqueous or polymeric. NPNCs are engineered for high-performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier pr...

Forest and co-workers report on p. 2029 that nematic polymer nanocomposite (NPNC) films can be processed in steady shear flows, which generate complex orientational distributions of the nanorod inclusions. Distribution functions for a benchmark NPNC (11 vol.-% of 1 nm × 200 nm rods) are computed for a range of shear rates, yielding a bifurcation di...

Composites made from two linear isotropic elastic materials are subjected to a uniform hydrostatic stress. It is assumed that only the volume fraction of each elastic material is known. Lower bounds on all rth moments of the hydrostatic stress field inside each phase are obtained for r⩾2. A lower bound on the maximum value of the hydrostatic stress...

Our aim here is to predict elongational flow-induced enhancements in thermal or electrical conductivity of liquid crystal polymer (LCP) nano-composites. To do so, we combine two classical mathematical asymptotic analyses: slender longwave hydro-thermo-dynamics for fibers and exact analysis of pure elongation of LCPs in solvents for bulk phases with...