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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2024

Scalable Multiphysics Block Preconditioning for Low Mach Number Compressible Resistive MHD with Application to Magnetic Confinement Fusion

This study investigates multiphysics block preconditioners that are critical in devising scalable Newton–Krylov iterative solvers for longer time-scale fully implicit fluid plasma models. The specific model of interest is the visco-resistive, low Mach number, compressible magnetohydrodynamics (MHD) model. This model describes the dynamics of conducting fluids in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics. The specific application of interest that motivates this study is the macroscopic simulation of longer time-scale stability and disruptions of magnetic confinement fusion devices, specifically the ITER Tokamak. The computational solution of the governing balance equations for mass, momentum, heat transfer, and magnetic induction for resistive MHD systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena as well as the significant range of time and length scales that the interactions of these physical mechanisms produce. To handle the range of time and spatial scales of interest, a fully implicit unstructured variational multiscale finite element formulation is employed. For the scalable solution of the Newton linearized systems, fully coupled block preconditioners are designed to leverage algebraic multigrid subsolves. In conclusion, results are presented for the strong and weak scaling of the method as well as the robustness of these techniques for a large range of Lundquist numbers.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A large deformation multiphase continuum mechanics model for shock loading of soft porous materials

A large deformation, coupled finite-element (FE) model is developed to simulate the multiphase response of soft porous materials subjected to high strain-rate loading. The approach is based on the theory of porous media (TPM) at large deformations. Simplifications to the one-dimensional regime studied in the numerical simulations follow. An overview of several different time integration schemes is presented for the purpose of solving the nonlinear dynamic coupled balance of momenta (mixture and fluid) and balance of mass of the mixture equations. Numerical examples are presented for (i) verification against closed-form analytical solutions assuming small loads, (ii) demonstrating large deformation effects at high strain-rate, and (iii) showing differences in deformations between a single-phase elastodynamics model with occluded compressible pore fluid and a multiphase poroelastodynamics model at high strain-rate. The multiphase model shows that the relative motion of the pore fluid significantly dampens the deformation response of the solid skeleton as compared to the single-phase model, and makes it possible to extract quantitative values for the stresses of the different constituents, thereby allowing one to form preliminary conclusions about the onset of damage in the solid skeleton. The novelty of the current work is developing a multiphase, large deformation, mixture theory numerical model for high strain-rate loading of soft porous materials. It was discovered that explicit, adaptive time-stepping Runge–Kutta schemes offer high accuracy at relatively low cost when compared to traditional implicit or explicit central difference time-stepping schemes for shock-like loadings. Here, shock viscosity is added to the mixture momentum balance equation to regularize the shock front, and a stabilization term is added to the mixture mass balance equation to stabilize equal order interpolation finite elements for the coupled finite element solution of multiphase materials.

Engineering↗
DOE OSTI2021

Ball-on-ring test validation for equibiaxial flexural strength testing of engineered ceramics

The validation of a ball-on-ring, equibiaxial flexural strength method to obtain the transverse rupture strength (TRS) of right cylindrical ceramic specimens was performed in this study. Validation of the test method was achieved using commercially available engineered high purity alumina disks and finite element (FE) model analysis. The validated fixture was then used to obtain the TRS and Weibull statistical analysis of MgO-partially stabilized zirconia (MSZ) and Y 2 O 3 -partially stabilized zirconia (YSZ) ceramic disks. TRS data for alumina, MSZ, and YSZ agreed with the TRS values reported in the literature. A statistically relevant number of samples (N > 30) for each material were tested to allow for a Weibull statistical analysis. Weibull parameters for these materials were within the expected values for engineered ceramics. The characteristic strength for alumina, MSZ, and YSZ were determined to be 289, 786, and 814 MPa, respectively. The Weibull modulus was determined between 10 and 25 for each material, which is typical of engineered ceramics. In addition, FE model results were in close agreement with experimental fracture values for the three ceramic materials tested in this study.

36 MATERIALS SCIENCE↗
DOE OSTI2021

A weighted Shifted Boundary Method for free surface flow problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. Finally, we present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Modeling of High-Temperature Corrosion of Zirconium Alloys Using the eXtended Finite Element Method (X-FEM)

Oxidation modeling in modern nuclear fuel performance codes is currently limited by the lack of coupling with mechanics, thus preventing proper description of how high-temperature oxidation impacts mechanical properties. This is mostly due to the fact that the finite difference formalism adopted in corrosion models is incompatible with the direct coupling with mechanics in the finite element modeling employed in modern nuclear fuel performance codes. In this study, a physically based zirconium alloy corrosion model called the Coupled-Current Charge Compensation (C4) model, which was initially developed for operating temperature conditions, has been updated to include high-temperature corrosion in order to provide additional critical information (e.g., oxygen concentration profile) under loss-of-coolant accident (LOCA) conditions—information lacking in existing empirical models. The C4 model was implemented in the MOOSE finite-element framework developed at Idaho National Laboratory, enabling it to be used in the BISON nuclear fuel performance code based on the MOOSE framework. To precisely track the different interfaces at a relatively low computational cost, the eXtended Finite Element Method (X-FEM) was applied in MOOSE. The model’s results were compared to those of existing empirical models as well as metallographic analysis of high-temperature oxidized Zircaloy-4 coupons. Oxygen diffusivities in the a and ß phases resulting from this comparison closely agree with those found in the literature. The C4 model implemented with X-FEM in MOOSE now has the capability to accurately predict oxide, oxygen-stabilized a, and prior ß phase layer growth kinetics under isothermal exposure at high temperature (1000–1500°C). Furthermore, in contrast with the empirical models, the C4 model accounts for the finite thickness of the fuel cladding. It can predict the oxygen concentration profile evolution through the whole cladding, enabling evaluation of the remaining ductile thickness—a crucial variable for modeling the mechanical behavior of the fuel cladding under LOCA. Furthermore, this implementation allows direct coupling with mechanics, at a low computing cost, using finite-element-based nuclear fuel performance codes such as BISON.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Development, Implementation, and Verification of Partially-Ionized Collisional Multifluid Plasma Models in Drekar

We consider the development of multifluid models for partially ionized multispecies plasmas. The models are composed of a standard set of five-moment fluid equations for each species plus a description of electromagnetics. The most general model considered utilizes a full set of fluid equations for each charge state of each atomic species, plus a set of fluid equations for electrons. The fluid equations are coupled through source terms describing electromagnetic coupling, ionization, recombination, charge exchange, and elastic scattering collisions in the low-density coronal limit. The form of each of these source terms is described in detail, and references for required rate coefficients are identified for a diverse range of atomic species. Initial efforts have been made to extend these models to incorporate some higher-density collisional effects, including ionization potential depression and three- body recombination. Some reductions of the general multifluid model are considered. First, a reduced multifluid model is derived which averages over all of the charge states (including neutrals) of each atomic species in the general multifluid model. The resulting model maintains full consistency with the general multifluid model from which it is derived by leveraging a quasi-steady-state collisional ionization equilibrium assumption to recover the ionization fractions required to make use of the general collision models. Further reductions are briefly considered to derive certain components of a single-fluid magnetohydrodynamics (MHD) model. In this case, a generalized Ohm's law is obtained, and the standard MHD resistivity is expressed in terms of the collisional models used in the general multifluid model. A number of numerical considerations required to obtain robust implementations of these multifluid models are discussed. First, an algebraic flux correction (AFC) stabilization approach for a continuous Galerkin finite element discretization of the multifluid system is described in which the characteristic speeds used in the stabilization of the fluid systems are synchronized across all species in the model. It is demonstrated that this synchronization is crucial in order to obtain a robust discretization of the multifluid system. Additionally, several different formulations are considered for describing the electromagnetics portion of the multifluid system using nodal continuous Galerkin finite element discretizations. The formulations considered include a parabolic divergence cleaning method and an implicit projection method for the traditional curl formulation of Maxwell's equations, a purely- hyperbolic potential-based formulation of Maxwell's equations, and a mixed hyperbolic-elliptic potential-based formulation of Maxwell's equations. Some advantages and disadvantages of each formulation are explored to compare solution robustness and the ease of use of each formulation. Numerical results are presented to demonstrate the accuracy and robustness of various components of our implementation. Analytic solutions for a spatially homogeneous damped plasma oscillation are derived in order to verify the implementation of the source terms for electromagnetic coupling and elastic collisions between fluid species. Ionization balance as a function of electron temperature is evaluated for several atomic species of interest by comparing to steady-state calculations using various sets of ionization and recombination rate coefficients. Several test problems in one and two spatial dimensions are used to demonstrate the accuracy and robustness of the discretization and stabilization approach for the fluid components of the multifluid system. This includes standard test problems for electrostatic and electromagnetic shock tubes in the two-fluid and ideal shock-MHD limits, a cylindrical diocotron instability, and the GEM challenge magnetic reconnection problem. A one-dimensional simplified prototype of an argon gas puff configuration as deployed on Sandia's Z-machine is used as a demonstration to exercise the full range of capabilities associated with the general multifluid model.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Islet: interpolation semi-Lagrangian element-based transport

Abstract. Advection of trace species, or tracers, also called tracer transport, in models of the atmosphere and other physical domains is an important and potentially computationally expensive part of a model's dynamical core. Semi-Lagrangian (SL) advection methods are efficient because they permit a time step much larger than the advective stability limit for explicit Eulerian methods without requiring the solution of a globally coupled system of equations as implicit Eulerian methods do. Thus, to reduce the computational expense of tracer transport, dynamical cores often use SL methods to advect tracers. The class of interpolation semi-Lagrangian (ISL) methods contains potentially extremely efficient SL methods. We describe a finite-element ISL transport method that we call the interpolation semi-Lagrangian element-based transport (Islet) method, such as for use with atmosphere models discretized using the spectral element method. The Islet method uses three grids that share an element grid: a dynamics grid supporting, for example, the Gauss–Legendre–Lobatto basis of degree three; a physics parameterizations grid with a configurable number of finite-volume subcells per element; and a tracer grid supporting use of Islet bases with particular basis again configurable. This method provides extremely accurate tracer transport and excellent diagnostic values in a number of verification problems.

58 GEOSCIENCES↗
DOE OSTI2024

A high-order Shifted Interface Method for Lagrangian shock hydrodynamics

Here, we present a new method for two-material Lagrangian hydrodynamics, which combines the Shifted Interface Method (SIM) with a high-order Finite Element Method. Our approach relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. The interface is represented by the zero level-set of a continuous high-order finite element function that moves with the material velocity. This strategy allows to evolve curved material interfaces inside curved elements. By reformulating the original interface problem over a surrogate (approximate) interface, located in proximity of the true interface, the SIM avoids cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. Accuracy is maintained by modifying the original interface conditions using Taylor expansions. We demonstrate the performance of the proposed algorithms on established numerical benchmarks in one, two and three dimensions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Optimization-based, property-preserving finite element methods for scalar advection equations and their connection to Algebraic Flux Correction

In this paper, we continue our efforts to exploit optimization and control ideas as a common foundation for the development of property-preserving numerical methods. Here we focus on a class of scalar advection equations whose solutions have fixed mass in a given Eulerian region and constant bounds in any Lagrangian volume. Our approach separates discretization of the equations from the preservation of their solution properties by treating the latter as optimization constraints. This relieves the discretization process from having to comply with additional restrictions and makes stability and accuracy the sole considerations in its design. A property-preserving solution is then sought as a state that minimizes the distance to an optimally accurate but not property-preserving target solution computed by the scheme, subject to constraints enforcing discrete proxies of the desired properties. Furthermore, we consider two such formulations in which the optimization variables are given by the nodal solution values and suitably defined nodal fluxes, respectively. A key result of the paper reveals that a standard Algebraic Flux Correction (AFC) scheme is a modified version of the second formulation obtained by shrinking its feasible set to a hypercube. In conclusion, we present numerical studies illustrating the optimization-based formulations and comparing them with AFC

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Sequential formulation of all‐way coupled finite strain thermoporomechanics for largely deformable gas hydrate deposits

We develop a numerically stable sequential formulation of thermoporomechanics for largely deformable gas hydrate deposits, extended from the fixed stress split of infinitesimal transformation. Constitutive equations are based on the total Lagrangian approach for both flow and geomechanics, including dynamic full tensor permeability and thermal conductivity updated from the deformation gradient. For space discretization, we take the cell-centered finite volume and node-based finite element method for flow and geomechanics, respectively. Then, we propose a sequential implicit method for all-way coupled thermoporomechanics, where the nonisothermal multiphase flow problem of gas hydrates is solved implicitly first and then the geomechanics problem is solved implicitly at the next step. During solution of the flow problem, we fix the rate of first Pioal total stress for numerical stability as well as apply porosity correction and entropy correction to account for geomechanical effects. We test numerical examples where flow and geomechanics parameters are based on deep oceanic gas hydrate deposits. When applying depressurization, even though the results between the infinitesimal transformation and finite strain geomechanics are similar in the early stages due to small deformation, we find differences between them in the late times as deformation becomes large. Accordingly, permeability and thermal conductivity tensors become nonisotropic full tensors although they are initially isotropic. Furthermore, we identify numerical stability of the developed sequential method from the test cases that exhibit the highly complex coupled gas hydrate systems with large deformation. Thus, the proposed sequential formulation can be applied in largely deformable gas hydrate systems.

42 ENGINEERING↗
DOE OSTI2021

A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners

Abstract Battery performance is strongly correlated with electrode microstructure. Electrode materials for lithium-ion batteries have complex microstructure geometries that require millions of degrees of freedom to solve the electrochemical system at the microstructure scale. A fast-iterative solver with an appropriate preconditioner is then required to simulate large representative volume in a reasonable time. In this work, a finite element electrochemical model is developed to resolve the concentration and potential within the electrode active materials and the electrolyte domains at the microstructure scale, with an emphasis on numerical stability and scaling performances. The block Gauss-Seidel (BGS) numerical method is implemented because the system of equations within the electrodes is coupled only through the nonlinear Butler–Volmer equation, which governs the electrochemical reaction at the interface between the domains. The best solution strategy found in this work consists of splitting the system into two blocks—one for the concentration and one for the potential field—and then performing block generalized minimal residual preconditioned with algebraic multigrid, using the FEniCS and the Portable, Extensible Toolkit for Scientific Computation libraries. Significant improvements in terms of time to solution (six times faster) and memory usage (halving) are achieved compared with the MUltifrontal Massively Parallel sparse direct Solver. Additionally, BGS experiences decent strong parallel scaling within the electrode domains. Last, the system of equations is modified to specifically address numerical instability induced by electrolyte depletion, which is particularly valuable for simulating fast-charge scenarios relevant for automotive application.

25 ENERGY STORAGE↗
DOE OSTI2022

The Schwarz Alternating Method for the Seamless Coupling of Nonlinear Reduced Order Models and Full Order Models

Projection-based model order reduction allows for the parsimonious representation of full order models (FOMs), typically obtained through the discretization of a set of partial differential equations (PDEs) using conventional techniques (e.g., finite element, finite volume, finite difference methods) where the discretization may contain a very large number of degrees of freedom. As a result of this more compact representation, the resulting projection-based reduced order models (ROMs) can achieve considerable computational speedups, which are especially useful in real-time or multi-query analyses. One known deficiency of projection-based ROMs is that they can suffer from a lack of robustness, stability and accuracy, especially in the predictive regime, which ultimately limits their useful application. Another research gap that has prevented the widespread adoption of ROMs within the modeling and simulation community is the lack of theoretical and algorithmic foundations necessary for the “plug-and-play” integration of these models into existing multi-scale and multi-physics frameworks. This paper describes a new methodology that has the potential to address both of the aforementioned deficiencies by coupling projection-based ROMs with each other as well as with conventional FOMs by means of the Schwarz alternating method [41]. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and concurrent multiscale coupling of finite element FOMs in solid mechanics [35, 36], we present a new extension of the Schwarz framework that enables FOM-ROM and ROM-ROM coupling, following a domain decomposition of the physical geometry on which a PDE is posed. In order to maintain efficiency and achieve computation speed-ups, we employ hyper-reduction via the Energy-Conserving Sampling and Weighting (ECSW) approach [13]. We evaluate the proposed coupling approach in the reproductive as well as in the predictive regime on a canonical test case that involves the dynamic propagation of a traveling wave in a nonlinear hyper-elastic material.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Analysis of the weighted shifted boundary method for the Poisson and Stokes problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.

Approximate domain boundaries↗
DOE OSTI2023

The Flux-Differencing Discontinuous Galerkin Method Applied to an Idealized Fully Compressible Nonhydrostatic Dry Atmosphere

Dynamical cores used to study the circulation of the atmosphere employ various numerical methods ranging from finite-volume, spectral element, global spectral, and hybrid methods. In this work, we explore the use of Flux-Differencing Discontinuous Galerkin (FDDG) methods to simulate a fully compressible dry atmosphere at various resolutions. We show that the method offers a judicious compromise between high-order accuracy and stability for large-eddy simulations and simulations of the atmospheric general circulation. In particular, filters, divergence damping, diffusion, hyperdiffusion, or sponge-layers are not required to ensure stability; only the numerical dissipation naturally afforded by FDDG is necessary. We apply the method to the simulation of dry convection in an atmospheric boundary layer and in a global atmospheric dynamical core in the standard benchmark of Held and Suarez

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines

Here, this paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an extended finite element method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

42 ENGINEERING↗
DOE OSTI2025

Analysis Background & Noise in Stretched Wire Alignment Technique Measurements

The Stretched-Wire Alignment Technique (SWAT) is one method of magnet alignment for linear induction accelerators. The applications of SWAT have been implemented for aligning solenoid magnets on the Scorpius linear induction accelerator which will be sited at the Nevada National Security Site and the Flash X-Ray (FXR) linear induction accelerator at Lawrence Livermore National Laboratory’s Contained Firing Facility. This article describes both systematic (repeatable) and random sources of background and noise as well as practical ways to eliminate or reduce them to acceptable levels. Systematic sources include reflections from wire ends, rapid sag due to ohmic heating of the wire, magnetic materials, and shot rate. Random sources include air currents, vibration of nearby equipment, mechanical stability of test equipment, and the instruments used to measure the wire motion. Mitigations include curve fitting and adaptive noise signal cancellation, and mechanical damping. Finite Element Analysis (FEA) was used to identify and resolve a repeatable wire vibration frequency interfering with the signal resolution. Two stretched wire alignment technique set ups from Sandia National Labs and Lawrence Livermore National Lab have shown background noise sources and ways of mitigating them by either analysis methods or change of mechanical configuration. Conclusions that were drawn included the severe sensitivity of the deflection to even small external interferences of the SWAT wire such that it requires attention to detail in mechanical set up and analysis.

Linear Inductive Accelerator↗
DOE OSTI2021

Emerging contact force heterogeneity in ordered soft granular media

Under external perturbations, inter-particle forces in disordered granular media are well known to form a heterogeneous distribution with filamentary patterns. Better understanding these forces and the distribution is important for predicting the collective behavior of granular media, the media second only to water as the most manipulated material in global industry. However, studies in this regard so far have been largely confined to granular media exhibiting only geometric heterogeneity, leaving the dimension of mechanical heterogeneity a rather uncharted area. In this work, through a FEM contact mechanics model, we show that a heterogeneous inter-particle force distribution can also emerge from the dimension of mechanical heterogeneity alone. Specifically, we numerically study inter-particle forces in packing of mechanically heterogeneous disks arranged over either a square or a hexagonal lattice and under quasi-static isotropic compression. Our results show that, at the system scale, a hexagonal packing exhibit a more heterogeneous inter-particle force distribution than a square packing does; At the particle scale, for both packing lattices, preliminary analysis shows the consistent coexistence of outliers (i.e., softer disks sustaining larger forces while stiffer disks sustaining smaller forces) in comparison to their homogeneous counterparts, which implies the existence of nonlocal effect. Further analysis on the portion of outliers and on spatial contact force correlations suggest that the hexagonal packing shows more pronounced nonlocal effect over the square packing under small mechanical heterogeneity. However, such trend is reversed when assemblies becomes more mechanically heterogeneous. Lastly, we confirm that, in the absence of particle reorganization events, contact friction merely plays the role of packing stabilization while its variation has little effect on inter-particle forces and their distribution.

36 MATERIALS SCIENCE↗