Continuum Domain Research Articles

Investigating fracture mechanisms in glassy polymers using coupled particle-continuum simulations

We study the fracture behavior of glassy polymers using a multiscale simulation method that integrates a molecular dynamics (MD) system within a continuum domain. By employing a nonlinear viscoelastic constitutive model in the continuum domain, the MD system undergoes non-uniform deformation with flexible boundaries through interaction with the surrounding continuum. Systems with pre-defined double cracks are subjected to tensile stretch under various geometric constraints and bond breakage criteria. The simulation results show that geometric constraints primarily affect deformation behavior at small strains, but their influence diminishes at larger strains. In the stage of fracture, increased deformation correlates with a narrowing distribution of microscopic structures at molecular scales, such as bond length. This distribution converges across different systems just before fracture, irrespective of ultimate stress, fracture strain, bond breakage criteria, and geometric constraints. This convergence suggests that the distribution of microscopic structures is a fundamental property linked to the fracture behavior of glassy polymers.

Efficient atomistic/continuum coupling using lattice Green’s functions

Coupled Atomistic/Continuum (A/C) simulation methods significantly reduce the number of degrees of freedom in the solution of a boundary value problem by using continuum elasticity in domains where defects interact elastically. A/C coupling involves two boundaries: (a) an inner boundary that connects the atomistic and continuum domain and (b) an outer boundary where displacement or force boundary conditions are applied. Here, the lattice Green’s function (LGF)–based flexible boundary condition method is combined with the LGF-based discrete boundary element method to satisfy coupling and boundary conditions at the inner and outer boundaries, respectively, of 3D domains. The coupling algorithms for Dirichlet boundary conditions are presented, with other cases being similar. The LGF is further replaced by a more efficient coarsened LGF method that coarsens the atomistic degrees of freedom at the outer boundary. The scaling of memory and computation versus 3D size and degree of coarsening are discussed. The method and its accuracy are demonstrated using a large 2D problem involving applied forces inside the atomistic domain.

High-performance mid-infrared refractive index sensing all-dielectric metasurface with dual resonance excited by bound states in the continuum and toroidal dipole mode

Converting bound states in the continuum (BICs) domain to quasi-BICs regime effectively achieves high Q-factor excitation, enabling refractive index sensing. In this paper, we propose an all-dielectric metasurface consisting of four silicon nanopore cylinders. While maintaining a four-fold symmetry, an asymmetric distribution of nanopores is introduced, creating two resonances in the mid-infrared range. Through near-field analysis and multipole decomposition, it is demonstrated that magnetic dipoles (MD) and toroidal dipoles (TD) dominate the resonance modes, and the sensing performance of the structure is investigated. The quality factor (Q) and sensitivity reach 6688 and 1140nm/RIU, respectively. The sensing characteristics in different refractive index (RI) media are analyzed, and the results show that the refractive index sensing capability of the dielectric metasurface is influenced by its structural parameters and exhibits a significant linear response relationship. Furthermore, by varying the structural parameters of the dielectric metasurface, we also discovered the differences in sensing performance within different wavelength ranges. We study all polarization angle robust quasi-BICs in the mid-infrared spectral region, which can provide new references for polarization-insensitive metasurfaces with dual resonances. Polarization-insensitive technology can avoid the need for compensating the polarization state of the light signal in the sensor and improve the performance and reliability of the sensor.

Multiscale modeling of nanoindentation and nanoscratching by generalized particle method

Most concurrent multiscale methods that expand atomistic region by continuum domain suffer from inconsistent material constitutive properties, which affect the integrity of model in the interface of atomistic and continuum domains. In this paper, the Generalized Particle (GP) method is employed to simulate nanoindentation and nanoscratching of a single-crystal aluminum sample. The main advantage of the GP method lies in its ability to extend the simulation model while maintaining consistent atomic properties across all scales. Coarsening of the atomic domain has been conducted through two-scale and three-scale GP model. The results showed a strong consistency between the results of full atomic simulations and those achieved through the GP method for both nanoindentation and nanoscratch simulations. Also, wave reflections were not seen at the interfaces. The study revealed that an increase in the number of distinct particle domains led to a reduction in the accuracy of multiscale simulations.

A Mechanical Picture of Fractal Darcy’s Law

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

The case for acceleration of prospective multi- stakeholder led community-based research in young Black women with triple negative breast cancer (TNBC).

1089 Background: Black women face a three- fold increase in TNBC BRCA1 gene mutations. In TNBC, there is a higher rate of recurrence and metastasis to other organs. Lower access to timely care and more advance stage at diagnosis contribute to poorer outcomes. Black women have the lowest survival rate at each stage of TNBC diagnosis compared to other demographic groups. Research Question: Over the past 10 years, what real world community interventions have been published regarding risk for and diagnosis with TNBC in Black or African American women? Methods: Using the following keywords: TNBC, diversity, health equity, young adults, Black, African American, authors conducted PUBMED and Google Scholar searches for years 2011 to 2022 to summarize publications on community-based interventions with Black or African American women at risk of or diagnosed with TNBC. Results: A validated publication assessment revealed 70 publications: 63 retrospective and 7 prospective studies. Prospective studies were categorized using the NASEM cancer continuum of care domains. The number of studies that addressed the domains included: risk=1, screening =1, diagnosis=3, treatment=2, survivorship=1 end of life=0, outcomes=1. No studies reported the following criteria: A co-creation approach with community-based participatory research principles; interventions led by patients and Black women who are at risk for TNBC; multi stakeholder interventions led by community leaders or patient organizations with long-standing trust of diverse populations; addressing and resolving social drivers of health inequities, mental health, palliative care services or implicit bias; or inclusion in clinical studies. Conclusions: It is critical to address the disproportionate impact of TNBC in partnership with Black women. To achieve equitable health outcomes, we advise public and private sector leaders to make more funding available to support community-engaged approaches in high prevalence geographic areas. 1) Provide CBPR training and opportunities for diverse investigators and patient- and community-based organizations. 2) Build capacity for social needs assessment and interventions. 3) Employ community-based implementation science focused on all care continuum domains including the gaps between domains. 4) Increase public awareness of health issues in Black women and address resistance, access and eligibility for participation in clinical research. 5) Offer RFPs to transform care through collaboration between researchers, providers, patient and community-based organizations, health departments, and payers; engaging clinical and lay navigators and community health workers; design solutions that address barriers; and development and implementation of policy with a focus on standardization, accountability, and enforceability.

Clues on the presence and segregation of very massive stars in the Sunburst Lyman-continuum cluster at z = 2.37

We report the identification of very massive stars (VMS; mass > 100 M⊙) that may be segregated in the center of the young massive star cluster at z = 2.37 hosted in the lensed galaxy called Sunburst galaxy. This result is based on two pieces of evidence: (1) VLT/MUSE spectra of several multiple images of the same star cluster show key spectral signatures of VMS, such as the He IIλ1640 broad emission, N IVλ1486 emission, and an N IVλ1720 P Cygni profile. In particular, He IIλ1640 is broad (∼1610 ± 300 km s−1), with an equivalent width of 3 Å, and asymmetric profile. These features require an extremely young (∼2.5 Myr) stellar population component in which the masses of the stars exceed 100 M⊙. When a Salpeter initial mass function and BPASS models for normal massive stars are assumed, the observed spectral features require ∼400 VMS. (2) The same star cluster is detected at a signal-to-noise ratio of ∼100 in the Lyman continuum domain (λ < 900 Å). The Lyman continuum emission emerges from a region with a radius that is at least twice smaller than what is observed at 1700 Å (independently of magnification) and is located in the center of the cluster. After delensing, the effective radii in absolute scales are Reff[LyC] ∼ 4.7 ± 1.5 pc and Reff[1700] = 7.8 ± 1.4 pc. The Lyman continuum radiation is mainly produced by hot and massive stars, which implies that their spatial distribution (including that of VMS) is preferentially more confined in the central parts of the cluster. Approximately 400 VMS hosted by a cluster of ∼107 M⊙ produce ∼15% of the escaping Lyman continuum photons, and the remaining photons are produced by other massive early-type stars.

Crack tip enhanced phase-field model for crack evolution in crystalline Ti6Al from concurrent crystal plasticity FE-molecular dynamics simulations

Dislocations nucleating from a crack-tip contribute to the plasticity evolution in the vicinity of crack, as well as to the crack propagation process. This paper systematically develops a method for enhancing the defect energy, accounting for crack-tip nucleated dislocations in the Helmholtz free energy density functional, associated with the phase-field formulation of crack evolution, in a coupled crystal plasticity finite element phase field (CPFE-PF) model. A concurrent crystal plasticity FE- hyper-dynamics accelerated molecular dynamics (CPFE-MD) model is created. The model that identifies, transfers, and propagates dislocations from the atomistic to continuum domain, is used to generate a dataset of crack tip dislocation density evolution along with different state variables. The paper focuses on crack tip plasticity mechanisms for the crystalline alloy Ti6Al. Using a Bayesian inference approach the critical state variables that affect the evolution of crack tip nucleated dislocation density are inferred. A functional form of the evolution of dislocation density in terms of the state variables is derived by employing a genetic programming based symbolic regression (GPSR) approach. The contribution of nucleated dislocation densities to effective plastic strain evolution at the crack tip is validated using the CPFE-MD model simulations. A comparison of the crack path and other state variables in the vicinity of the crack with and without contributions from the nucleated dislocations shows the importance of this augmentation on crack evolution.

An optimal transport problem with storage fees

We establish basic properties of a variant of the semi-discrete optimal transport problem in a relatively general setting. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support of the target measure, and a “storage fee” function. The goal is to find a map for which the total transport cost plus the storage fee evaluated on the masses of the pushforward of the source measure is minimized. We prove existence and uniqueness for the problem, derive a dual problem for which strong duality holds, and give a characterization of dual maximizers and primal minimizers. Additionally, we find some stability results for minimizers and a Γ-convergence result as the target set becomes denser and denser in a continuum domain.

A particle‐continuum coupling method for amorphous polymers with multiple particle‐based domains

AbstractThis contribution presents a partitioned‐domain particle‐continuum coupling method for amorphous polymers with multiple particle‐based domains. The coupling method treats the particle‐based domains with molecular dynamics (MD) simulations and the continuum domain discretized by the Finite Element (FE) method. In the continuum domain, a viscoelastic‐viscoplastic (VE‐VP) constitutive model derived from MD simulation results of the polymer at molecular resolution is employed. The effects of the minimum distances between the domains, the distribution and the number of the MD domains as well as the strain rates are studied under uniaxial tension. This method is a precursor for multiscale simulations of polymer‐based nanocomposites (PNC).

Quantum Topology Optimization via Quantum Annealing

We present a quantum annealing-based solution method for topology optimization (TO). In particular, we consider TO in a more general setting, i.e., applied to structures of continuum domains where designs are represented as distributed functions, referred to as continuum TO problems. According to the problem's properties and structure, we formulate appropriate sub-problems that can be solved on an annealing-based quantum computer. The methodology established can effectively tackle continuum TO problems formulated as mixed-integer nonlinear programs. To maintain the resulting sub-problems small enough to be solved on quantum computers currently accessible with small numbers of qubits and limited connectivity, we further develop a splitting approach that splits the problem into two parts: the first part can be efficiently solved on classical computers, and the second part with a reduced number of variables is solved on a quantum computer. By such, a practical continuum TO problem of varying scales can be handled on the D-Wave quantum annealer. More specifically, we concern the minimum compliance, a canonical TO problem that seeks an optimal distribution of materials to minimize the compliance with desired material usage. The superior performance of the developed methodology is assessed and compared with the stateof-the-art heuristic classical methods, in terms of both solution quality and computational efficiency. The present work hence provides a promising new avenue of applying quantum computing to practical designs of topology for various applications.

Slit-Strip Ising Boundary Conformal Field Theory 1: Discrete and Continuous Function Spaces

This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lattice domains. We find distinguished sets of functions characterized by their singular behavior in the three infinite directions in the slit-strip domains, and note in particular that natural subsets of these functions span analogues of Hardy spaces. We prove convergence results of the distinguished discrete holomorphic functions to the continuum ones. In the subsequent articles, the discrete holomorphic functions will be used for the calculation of the Ising model fusion coefficients (as well as for the diagonalization of the Ising transfer matrix), and the convergence of the functions is used to prove the convergence of the fusion coefficients. It will also be shown that the vertex operator algebra of the boundary conformal field theory can be recovered from the limit of the fusion coefficients via geometric transformations involving the distinguished continuum functions.

Finite element method–discrete element method bridging coupling for the modeling of gouge

AbstractWe discuss the multiscale modeling of a granular material trapped between continuum elastic domains. The amorphous granular region, usually termed “gouge,” is under high confinement pressure, to represent the loading of faults at depth. We model the granularity of gouge using the discrete element method (DEM), while the elastic regions surrounding it are represented with two continuum domains modeled with the finite element method (FEM). We resort to a concurrent coupling of the discrete and continuum domains for a proper transmission of waves between the discrete and continuum domains. The confinement pressure results in the appearance of a new kind of ghost forces, which we address via two different overlapping coupling strategies. The first one is a generalization to granular materials of the bridging method, which was originally introduced to couple continuum domains to regular atomic lattices. This method imposes a strong formulation for the Lagrange constraints at the coupling interface. The second strategy considers a weak formulation. Different DEM samples sizes are tested in order to determine at which scale a convergence of the elastic properties is reached. This scale sets the minimal mesh element size in the DEM/FEM interface necessary to avoid undesirable effects due to an elastic properties mismatch. Then, the two DEM/FEM strategies are compared for a system initially at equilibrium. While the performance of both strategies is adequate, we show that the strong coupling is the most stable one as it generates the least spurious numerical noise. Finally, as a practical example for the strong coupling approach, we analyze the propagation of pressure and shear waves through the FEM/DEM interface and discuss dispersion as function of the incoming wave frequency.

Predicting the elasto-plastic response of short fiber reinforced composites using a computationally efficient multi-scale framework based on physical matrix properties

Predicting the nonlinear mechanical response of short fiber reinforced composites (SFRCs) is a crucial and challenging task. In this paper, a computationally efficient multi-scale strategy is proposed to predict the anisotropic elasto-plastic behavior of SFRCs using the intrinsic mechanical behavior of the pure polymer and fibers without the requirements for reverse engineering. In doing so, different simple unit cells are first examined to find the one that can adequately describe the nonlinear mechanical response of SFRCs' representative volume elements (RVE) with aligned fibers. Considering the effects of packing configuration, fiber aspect ratio, volume fraction and material properties, the performance of different unit cells is investigated. Then, the homogenized mechanical responses of unit cells are linked to Hill's anisotropic plasticity model to correlate the mechanical response of the suggested unit cell to the continuum domain. Using the pseudo-grain approach and a numerical orientation averaging framework, the effects of fiber misalignment are taken into account. A multi-step homogenization strategy is also employed to consider the variation of fiber orientation tensor and volume fraction through the thickness. Finally, the validity and robustness of the proposed multi-scale strategy are extensively investigated based on the RVE-generated results and the available experimental observations.

A study by the lattice discrete element method for exploring the fractal nature of scale effects

Nowadays, there are many applications in the field of Engineering related to quasi-brittle materials such as ceramics, natural stones, and concrete, among others. When damage is produced, two phenomena can take place: the damage produced governs the collapse process when working with this type of material, and its random nature rules the nonlinear behavior up to the collapse. The interaction among clouds of micro-cracks generates the localization process that implies transforming a continuum domain into a discontinue one. This process also governs the size effect, that is, the changes of the global parameters as the strength and characteristic strain and energies when the size of the structure changes. Some aspects of the scaling law based on the fractal concepts proposed by Prof Carpinteri are analyzed in this work. On the other hand, the Discrete Method is an interesting option to be used in the simulation collapse process of quasi-brittle materials. This method can allow failures with relative ease. Moreover, it can also help to relax the continuum hypothesis. In the present work, a version of the Discrete Element Method is used to simulate the mechanical behavior of different size specimens until collapse by analyzing the size effect represented by this method. This work presents two sets of examples. Its results allow the researchers to see the connection between the numerical results regarding the size effect and the theoretical law based on the fractal dimension of the parameter studied. Two main aspects appear as a result of the analysis presented here. Understand better some aspects of the size effect using the numerical tool and show that the Lattice Discrete Element Method has enough robustness to be applied in the nonlinear analysis of structures built by quasi-brittle materials.

3D hybrid coupled dual continuum and discrete fracture model for simulation of CO2 injection into stimulated coal reservoirs with parallel implementation

This work presents the development of a 3D hybrid coupled dual continuum and discrete fracture model for simulating coupled flow, reaction and deformation processes relevant to fractured reservoirs with multiscale fracture system, e.g., coal and shale, efficiently and accurately. In this hybrid model, the natural fracture network and coal matrix are described together by a dual continuum approach and the large fractures are represented explicitly by the discrete fracture approach. A combination of different types of elements is used for spatial discretization. Large fractures are discretised with lower-dimensional interface elements and continuum domains with higher-dimensional elements. The coupling between the two models is achieved via the principle of superposition. To reduce computational time of simulations for complex and large-scale problems, a hybrid MPI/OpenMP parallel scheme is implemented in this work. The developed model is applied to investigate coupled thermal, hydraulic, and mechanical processes associated with CO2 sequestration and enhanced coalbed methane recovery. The results demonstrate capabilities of the model to adequately capture the effects of multiscale fracture system and their coupled behaviour during CO2 injection and methane recovery from coal reservoirs. Performance of the proposed parallelisation scheme was tested by comparing computation times of serial and parallel implementations. A good performance improvement was achieved, the speedup using parallelized scheme reaches up to about 10 times along with satisfactory scalability for considered application example. The findings of this work support developments and improvements of efficient advanced numerical models to study coupled THCM behaviour in fractured porous geomaterials.

Uniform convergence rates for Lipschitz learning on graphs

Abstract Lipschitz learning is a graph-based semisupervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence rates for solutions of the graph infinity Laplace equation as the number of vertices grows to infinity. Their continuum limits are absolutely minimizing Lipschitz extensions (AMLEs) with respect to the geodesic metric of the domain where the graph vertices are sampled from. We work under very general assumptions on the graph weights, the set of labeled vertices and the continuum domain. Our main contribution is that we obtain quantitative convergence rates even for very sparsely connected graphs, as they typically appear in applications like semisupervised learning. In particular, our framework allows for graph bandwidths down to the connectivity radius. For proving this we first show a quantitative convergence statement for graph distance functions to geodesic distance functions in the continuum. Using the ‘comparison with distance functions’ principle, we can pass these convergence statements to infinity harmonic functions and AMLEs.

Three-dimensional multiscale modeling of nanoindentation

Concurrent multiscale methods have been developed to reduce the degrees of freedom and reduce the effects of boundary conditions on the results of atomic simulations. In this paper, two simplified concurrent multiscale methods, one with handshake region and another without handshake region are used to investigate the nanoindentation process on a single crystal of Al at room temperature. The multiscale models are validated by observing reasonably well similarities in the load–depth curves obtained from multiscale and full MD simulations. Refining the element size down to atomic spacing resulted in high computational efforts while the analysis results do not improve significantly. Also, it is shown that by defining the thermostat in the atomistic part, wave reflections are eliminated at the interface of atomic and continuum domains. It is shown that by selecting appropriate dimensions of the atomic domain, there is no need to use nonlinear elasticity in the continuum region. Also, hardness is more affected by sample size than the elastic modulus.

A coarse-grained concurrent multiscale method for simulating brittle fracture

This paper proposed a concurrent multiscale method for modeling crack propagation in brittle solids. The method couples the continuum finite element domain with a recently-developed coarse-grained atomistic potential, which has been developed to simulate crack nucleation and propagation within the molecular dynamics domain. The main advantage of this potential is to suppress dislocation nucleation from the crack tip which significantly eases the coupling procedure. In this multiscale framework, complexities like large deformation, crack nucleation and propagation are simulated in the atomistic domain while the rest such as elastic wave propagation and boundary conditions are modeled in the continuum domain. As a result, the atomistic area will be limited around the cracking zone which increases the computational efficiency. Two examples of edge crack propagation and sliding surfaces in contact are presented and compared with the fully atomistic models to verify and check the efficiency of the proposed multiscale method. The comparison shows that the proposed method can accurately capture the crack formation while the computation cost reduces significantly.

Bayesian Optimization with Expensive Integrands

We propose a Bayesian optimization algorithm for objective functions that are sums or integrals of expensive-to-evaluate functions, allowing noisy evaluations. These objective functions arise in multi-task Bayesian optimization for tuning machine learning hyperparameters, optimization via simulation, and sequential design of experiments with random environmental conditions. Our method is average-case optimal by construction when a single evaluation of the integrand remains within our evaluation budget. Achieving this one-step optimality requires solving a challenging value of information optimization problem, for which we provide a novel efficient discretization-free computational method. We also provide consistency proofs for our method in both continuum and discrete finite domains for objective functions that are sums. In numerical experiments comparing against previous state-of-the-art methods, including those that also leverage sum or integral structure, our method performs as well or better across a wide range of problems and offers significant improvements when evaluations are noisy or the integrand varies smoothly in the integrated variables.