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Inspired by the gradient flow viewpoint of the Landau equation and corresponding dynamic formulation of the Landau metric in [arXiv:2007.08591], we develop a novel implicit particle method for the Landau equation in the framework of the JKO scheme. We first reformulate the Landau metric in a computationally friendly form, and then translate it into the Lagrangian viewpoint using the flow map. A key observation is that, while the flow map evolves according to a rather complicated integral equation, the unknown component is merely a score function of the corresponding density plus an additional term in the null space of the collision kernel. This insight guides us in designing and training the neural network for the flow map. Additionally, the objective function is in a double summation form, making it highly suitable for stochastic methods. Consequently, we design a tailored version of stochastic gradient descent that maintains particle interactions and significantly reduces the computational complexity. Compared to other deterministic particle methods, the proposed method enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.

We study a deterministic particle scheme to solve a scalar balance equation with nonlocal interaction and nonlinear mobility used to model congested dynamics. The main novelty with respect to "Radici-Stra [SIAM J. Math. Anal. 55.3 (2023)]" is the presence of a source term; this causes the solutions to no longer be probability measures, thus requiring a suitable adaptation of the numerical scheme and of the estimates leading to compactness., Comment: 26 pages, 1 figure

During the past decades, the numerical methods based on Navier-Stokes (N-S) equations and direct simulation Monte Carlo (DSMC) methods have been proved effective in simulating flows in the continuum and rarefied regimes, respectively. However, as single-scale methods, they face challenges in addressing common multi-scale problems, which are essential to simulate hypersonic flows around near-space vehicles and the flows in the micro-electro-mechanical systems. Hence, there is an urgent need for a method to predict multi-scale flows. In this work, a quantified model-competition (QMC) mechanism for diatomic multi-scale flows is derived from the integral solution of the Rykov model equations. This mechanism encapsulates both continuum and rarefied behaviors in a cell, weighted according to its local physical scale. By building upon the QMC mechanism, the N-S solver and DSMC solver are directly integrated within a cell to devise a simplified unified wave-particle (SUWP) method for diatomic gases. Specifically, the two-temperature equations considering the rotational energy are introduced into the kinetic inviscid flux (KIF) scheme and the N-S solver. As to the particle part, the collisionless DSMC solver is utilized to describe the non-equilibrium phenomenon. The proposed SUWP method for diatomic gases undergoes validation across a series of cases, including zero-dimensional homogeneous gas relaxation, one-dimensional normal shock structure, two-dimensional flow around the flat and the cylinder, and three-dimensional flows past the sphere and the blunt cone. Additionally, the implementation details of multi-scale wave-particle methods analysis and discussion are also undertaken in this work.

We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We introduce a new component in the local reconstruction step of the algorithm and demonstrate numerically that the modification can improve computational accuracy when a large curvature region is developed during evolution. The method also incorporates a recently developed constrained least-squares ghost sample points (CLS-GSP) formulation, which can lead to a better-conditioned discretized matrix for computing some surface differential operators. The proposed framework can incorporate many methods and link various approaches to the same problem. Several numerical experiments are carried out to show the accuracy and effectiveness of the proposed method.

We present a thorough analysis of computing the Generalized Alignment Index (GALI), a rapid and effective chaos indicator, through a simple multi-particle approach, which avoids the use of variational equations. We develop a theoretical leading-order estimation of the error in the computed GALI for both the variational method (VM) and the multi-particle method (MPM), and confirm its predictions through extensive numerical simulations of two well-known Hamiltonian models: the H\'enon-Heiles and the $\beta$-Fermi-Pasta-Ulam-Tsingou systems. For these models the GALIs of several orders are computed and the MPM results are compared to the VM outcomes. The dependence of the accuracy of the MPM on the renormalization time, integration time step, as well as the deviation vector size, is studied in detail. We find that the implementation if the MPM in double machine precision ($\varepsilon \approx 10^{-16}$) is reliable for deviation vector magnitudes centred around $d_0\approx \varepsilon^{1/2}$, renormalization times $\tau \lesssim 1$, and relative energy errors $E_r \lesssim \varepsilon^{1/2}$. These results are valid for systems with many degrees of freedom and for several orders of the GALIs, with the MPM particularly capturing very accurately the $\textrm{GALI}_2$ behavior. Our results show that the computation of the GALIs by the MPM is a robust and efficient method for investigating the global chaotic dynamics of autonomous Hamiltonian systems, something which is of distinct importance in cases where it is difficult to explicitly write the system's variational equation or when these equations are too cumbersome., Comment: 17 pages, 13 figures

The unified gas-kinetic wave-particle (UGKWP) method is a hybrid method for multiscale flow simulations, in which the contributions to the whole gas evolution from deterministic hydrodynamic wave and stochastic particle transport are combined simultaneously. Originally, the UGKWP method was developed as a direct modeling approach at discrete level. In this work, we revisit the time evolution of each part of the involved simulation particles and wave molecules in UGKWP, and present the corresponding kinetic equations. The resultant kinetic system can be viewed as a collision decomposition of the original kinetic equation, which can serve as a basis for developing other kinetic methods for flows in all flow regimes., Comment: 16 pages, 3 figures

The direct simulation Monte Carlo (DSMC) method is a widely used stochastic particle approach to solving the Boltzmann equation. However, its computational cost remains a major drawback, which can be attributed to statistical errors when handling flows with low Mach numbers. Thus, many studies have focused on variance reduction to reduce the computational cost. One approach is the deviational particle (DP) method, which focuses solely on modeling deviations from the equilibrium state. The DP method has been implemented in the low-variance deviational simulation Monte Carlo (LVDSMC) method, which has proven effective for monatomic gas simulations but faces limitations when extended to polyatomic gases. In this study, we present a practical DP method for polyatomic gas simulations that combines the LVDSMC method with the Larsen-Borgnakke (LB) model, which introduces a group reduction algorithm for the inelastic collision process. Numerical experiments demonstrated that the proposed method efficiently and accurately simulates flows across a relatively broad range of non-equilibrium values. Remarkably, the variance was reduced to about 5% that of the DSMC method.

The Unified Gas-Kinetic Wave-Particle (UGKWP) method is constructed for partially ionized plasma (PIP). This method possesses both multiscale and unified preserving (UP) properties. The multiscale property allows the method to capture a wide range of plasma physics, from the particle transport in the kinetic regime to the two-fluid and magnetohydrodynamics (MHD) in the near continuum regimes, with the variation of local cell Knudsen number and normalized Larmor radius.The unified preserving property ensures that the numerical time step is not limited by the particle collision time in the continuum regime for the capturing of dissipative macroscopic solutions of the resistivity, Hall-effect, and all the way to the ideal MHD equations.The UGKWP is clearly distinguishable from the classical single scale Particle-in-Cell/Monte Carlo Collision (PIC/MCC) methods.The UGKWP method combines the evolution of microscopic velocity distribution with the evolution of macroscopic mean field quantities, granting it UP properties. Moreover, the time step in UGKWP is not constrained by the plasma cyclotron period through the Crank-Nicolson scheme for fluid and electromagnetic field interactions. The momentum and energy exchange between different species is approximated by the Andries-Aoki-Perthame (AAP) model. Overall, the UGKWP method enables a smooth transition from the PIC method in the rarefied regime to the MHD solvers in the continuum regime. This method has been extensively tested on a variety of phenomena ranging from kinetic Landau damping to the macroscopic flow problems, such as the Brio-Wu shock tube, Orszag-Tang vortex, and Geospace Environmental Modeling (GEM) magnetic reconnection. These tests demonstrate that the proposed method can capture the fundamental features of PIP across different scales seamlessly.

We propose a novel score-based particle method for solving the Landau equation in plasmas, that seamlessly integrates learning with structure-preserving particle methods [arXiv:1910.03080]. Building upon the Lagrangian viewpoint of the Landau equation, a central challenge stems from the nonlinear dependence of the velocity field on the density. Our primary innovation lies in recognizing that this nonlinearity is in the form of the score function, which can be approximated dynamically via techniques from score-matching. The resulting method inherits the conservation properties of the deterministic particle method while sidestepping the necessity for kernel density estimation in [arXiv:1910.03080]. This streamlines computation and enhances scalability with dimensionality. Furthermore, we provide a theoretical estimate by demonstrating that the KL divergence between our approximation and the true solution can be effectively controlled by the score-matching loss. Additionally, by adopting the flow map viewpoint, we derive an update formula for exact density computation. Extensive examples have been provided to show the efficiency of the method, including a physically relevant case of Coulomb interaction.

We study an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $\lambda$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size, and we show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, suitable for the IPM. The IPM applies naturally to problems in unbounded domains, scales easily to high dimensions, and adapts to singular behaviors in the vanishing-noise limit. We show numerical examples in dimensions up to 16. The numerical results show that our numerical approximation of $\lambda$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

In this work, the fast-convolving reproducing kernel particle method (FC-RKPM) is introduced. This method is hundreds to millions of times faster than the traditional RKPM for 3D meshfree simulations. In this approach, the meshfree discretizations with RK approximation are expressed in terms of convolution sums. Fast Fourier transform (FFT) is then used to efficiently compute the convolutions. Certain modifications to the domain and shape functions are considered to maintain generality for complex geometries and arbitrary boundary conditions. The new method does not need to identify, store, and loop over the neighbors which is one of the bottleneck of the traditional meshfree methods. As a result, the run-times and memory allocations are independent of the number of neighbors and the shape function support size. As a model problem, the method is laid out for a Galerkin weak form of the Poisson problem with the RK approximation, and is verified in 1D, 2D, and 3D. Tables with run-times and allocated memory are presented to compare the performance of FC-RKPM with the traditional method in 3D. The performance is studied for various node numbers, support size, and approximation degree. All the implementation details and the roadmap for software development are also provided. Application of the new method to nonlinear and explicit problems are briefly discussed as well.

Chemotaxis systems play a crucial role in modeling the dynamics of bacterial and cellular behaviors, including propagation, aggregation, and pattern formation, all under the influence of chemical signals. One notable characteristic of these systems is their ability to simulate concentration phenomena, where cell density undergoes rapid growth near specific concentration points or along certain curves. Such growth can result in singular, spiky structures and lead to finite-time blowups. Our investigation focuses on the dynamics of the Patlak-Keller-Segel chemotaxis system and its two-species extensions. In the latter case, different species may exhibit distinct chemotactic sensitivities, giving rise to very different rates of cell density growth. Such a situation may be extremely challenging for numerical methods as they may fail to accurately capture the blowup of the slower-growing species mainly due to excessive numerical dissipation. In this paper, we propose a hybrid finite-difference-particle (FDP) method, in which a particle method is used to solve the chemotaxis equation(s), while finite difference schemes are employed to solve the chemoattractant equation. Thanks to the low-dissipation nature of the particle method, the proposed hybrid scheme is particularly adept at capturing the blowup behaviors in both one- and two-species cases. The proposed hybrid FDP methods are tested on a series of challenging examples, and the obtained numerical results demonstrate that our hybrid method can provide sharp resolution of the singular structures even with a relatively small number of particles. Moreover, in the two-species case, our method adeptly captures the blowing-up solution for the component with lower chemotactic sensitivity, a feature not observed in other works.

The multi-frequency radiation transport equation (RTE) system models the photon transport and the energy exchange process between the background material and different frequency photons. In this paper, the unified gas-kinetic wave-particle (UGKWP) method for multi-frequency RTE is developed to capture the multiscale non-equilibrium transport in different optical regimes. In the UGKWP, a multiscale evolution process is properly designed to obtain both non-equilibrium transport in the optically thin regime and thermal diffusion process in the optically thick regime automatically. At the same time, the coupled macroscopic energy equations for the photon and material are solved implicitly by the matrix-free source iteration method. With the wave-particle decomposition strategy, the UGKWP method has a dynamic adaptivity for different regime physics. In the optically thick regime, no particles will be sampled in the computational domain and the thermal diffusion solution for the photon transport will be recovered. While in the optically thin regime, stochastic particles will play a dominant role in the evolution and the non-equilibrium free transport of photon is automatically followed. In the frequency-dependent transport, the frequency carried by the simulating particle will be determined by a linear-frequency sampling strategy. In addition, to better resolve the sharp of opacity in the photon transport across a cell interface, the free streaming time of simulating particle in the UGKWP method will be reset when it passes through the interface. Moreover, the numerical relaxation time is properly defined to increase the particle proportion in the sharp opacity transition region in order to avoid numerical oscillation. Several typical test cases for the RTE system have been calculated to demonstrate the accuracy and reliability of the current frequency-dependent UGKWP method.

The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method arXiv:1910.03080 has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies BKW solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples., Comment: 25 pages, 31 figures

Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7-D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy.

This paper presents a comprehensive investigation into the modeling of rotor wake velocities, in a simplistic manner, using the Viscous Vortex Particle Method (VVPM). The study aims to accurately simulate wind velocities in the wake of helicopter rotors while comparing the VVPM with other established methods such as Momentum theory, Blade element theory, and the Free wake method. The VVPM is particularly promising due to its ability to simulate the interaction of rotor blade vorticities with a natural and efficient approach. The method employs a grid-free Lagrangian formulation, eliminating artificial numerical dissipation in the solution. To demonstrate the efficacy of the method, a Python code has been developed, enabling the visualisation of wake velocities for basic scenarios.

In deep geological repositories for high level nuclear waste with close canister spacings, bentonite buffers can experience temperatures higher than 100 {\deg}C. In this range of extreme temperatures, phenomenological constitutive laws face limitations in capturing the thermo-hydro-mechanical (THM) behavior of the bentonite, since the pre-defined functional constitutive laws often lack generality and flexibility to capture a wide range of complex coupling phenomena as well as the effects of stress state and path dependency. In this work, a deep neural network (DNN)-based soil-water retention curve (SWRC) of bentonite is introduced and integrated into a Reproducing Kernel Particle Method (RKPM) for conducting THM simulations of the bentonite buffer. The DNN-SWRC model incorporates temperature as an additional input variable, allowing it to learn the relationship between suction and degree of saturation under the general non-isothermal condition, which is difficult to represent using a phenomenological SWRC. For effective modeling of the tank-scale test, new axisymmetric Reproducing Kernel basis functions enriched with singular Dirichlet enforcement representing heater placement and an effective convective heat transfer coefficient representing thin-layer composite tank construction are developed. The proposed method is demonstrated through the modeling of a tank-scale experiment involving a cylindrical layer of MX-80 bentonite exposed to central heating., Comment: 51 pages, 19 figures

Nonequilibrium flows are common in aerospace engineering, and numerical simulations are vital in understanding non-equilibrium flow dynamics in spacecraft flight. The unified gas-kinetic wave-particle (UGKWP) method has been developed for multi-scale flow simulation, which models the coupled particle transport and collision within a numerical time in the flux evaluation across a cell interface. The UGKWP balances precision and efficiency in multiscale flow simulations, particularly in high-speed flow. In this study, the UGKWP method is used to simulate supersonic flow around a sphere, hypersonic flow around a space vehicle, nozzle plume into a vacuum, and side-jet impingement on hypersonic flow. The UGKWP accurately renders the Navier-Stokes solution within the nozzle, extending to free molecular flow in the external environment, all within a singular computation. Complicated structures in flow interaction have been captured by the UGKWP method. All simulation results have been verified through experimental measurements or Direct Simulation Monte Carlo (DSMC) methods. The UGKWP method requires only 60 GiB of memory to simulate a three-dimensional space vehicle with 560593 cells under different flow conditions, making it efficient and accurate for aerospace engineering applications., Comment: 39 pages with 31 figures and 13 tables

We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed., Comment: 36 pages, 17 figures, planning to submit for SIAM/ASA Journal on Uncertainty Quantification

This work presents an approach for automating the discretization and approximation procedures in constructing digital representations of composites from Micro-CT images featuring intricate microstructures. The proposed method is guided by the Support Vector Machine (SVM) classification, offering an effective approach for discretizing microstructural images. An SVM soft margin training process is introduced as a classification of heterogeneous material points, and image segmentation is accomplished by identifying support vectors through a local regularized optimization problem. In addition, an Interface-Modified Reproducing Kernel Particle Method (IM-RKPM) is proposed for appropriate approximations of weak discontinuities across material interfaces. The proposed method modifies the smooth kernel functions with a regularized heavy-side function concerning the material interfaces to alleviate Gibb's oscillations. This IM-RKPM is formulated without introducing duplicated degrees of freedom associated with the interface nodes commonly needed in the conventional treatments of weak discontinuities in the meshfree methods. Moreover, IM-RKPM can be implemented with various domain integration techniques, such as Stabilized Conforming Nodal Integration (SCNI). The extension of the proposed method to 3-dimension is straightforward, and the effectiveness of the proposed method is validated through the image-based modeling of polymer-ceramic composite microstructures., Comment: 58 pages, 51 figures, keywords: image-based modeling, support vector machine, reproducing kernel particle method, weak discontinuity, microstructures

The well-posedness of a non-local advection-selection-mutation problem deriving from adaptive dynamics models is shown for a wide family of initial data. A particle method is then developed, in order to approximate the solution of such problem by a regularised sum of weighted Dirac masses whose characteristics solve a suitably defined ODE system. The convergence of the particle method over any finite interval is shown and an explicit rate of convergence is given. Furthermore, we investigate the asymptotic-preserving properties of the method in large times, providing sufficient conditions for it to hold true as well as examples and counter-examples. Finally, we illustrate the method in two cases taken from the literature.

In this work, the Navier-Stokes (NS) solver is combined with the Direct simulation Monte Carlo (DSMC) solver in a direct way, under the wave-particle formulation [J. Comput. Phys. 401, 108977 (2020)]. Different from the classical domain decomposition method with buffer zone for overlap, in the proposed direct unified wave-particle (DUWP) method, the NS solver is coupled with DSMC solver on the level of algorithm. Automatically, in the rarefied flow regime, the DSMC solver leads the simulation, while the NS solver leads the continuum flow simulation. Thus advantages of accuracy and efficiency are both taken. At internal flow regimes, like the transition flow regime, the method is accurate as well because a kind of mesoscopic modeling is proposed in this work, which gives the DUWP method the multi-scale property. Specifically, as to the collision process, at $t < \tau$, it is supposed that only single collision happens, and the collision term of DSMC is just used. At $t > \tau$, it is derived that $1-\tau/\Delta t$ of particles should experience multiple collisions, which will be absorbed into the wave part and calculated by the NS solver. Then the DSMC and NS solver can be coupled in a direct and simple way, bringing about multi-scale property. The governing equation is derived and named as multi-scale Boltzmann equation. Different from the original wave-particle method, in the proposed DUWP method, the wave-particle formulation is no more restricted by the Boltzmann-BGK type model and the enormous research findings of DSMC and NS solvers can be utilized into much more complicated flows, like the thermochemical non-equilibrium flow. In this work, one-dimensional cases in monatomic argon gas are preliminarily tested, such as shock structures and Sod shock tubes.

We formulate a class of velocity-free finite-particle methods for mass transport problems based on a time-discrete incremental variational principle that combines entropy and the cost of particle transport, as measured by the Wasserstein metric. The incremental functional is further spatially discretized into finite particles, i.e., particles characterized by a fixed spatial profile of finite width, each carrying a fixed amount of mass. The motion of the particles is then governed by a competition between the cost of transport, that aims to keep the particles fixed, and entropy maximization, that aims to spread the particles so as to increase the entropy of the system. We show how the optimal width of the particles can be determined variationally by minimization of the governing incremental functional. Using this variational principle, we derive optimal scaling relations between the width of the particles, their number and the size of the domain. We also address matters of implementation including the acceleration of the computation of diffusive forces by exploiting the Gaussian decay of the particle profiles and by instituting fast nearest-neighbor searches. We demonstrate the robustness and versatility of the finite-particle method by means of a test problem concerned with the injection of mass into a sphere. There test results demonstrate the meshless character of the method in any spatial dimension, its ability to redistribute mass particles and follow their evolution in time, its ability to satisfy flux boundary conditions for general domains based solely on a distance function, and its robust convergence characteristics.

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche's method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.