Block-structured Adaptive Mesh Refinement Research Articles

Numerical investigation of the interaction between an interface and a decaying Lamb–Oseen vortex

The present study investigates the dynamics of the interface in the presence of a decaying Lamb–Oseen vortex, and four distinct wave patterns are observed: non-breaking waves with small periodic oscillations, plunging breakers, depression breakers, and gravity–capillary waves. The deformation of the interface is induced by a two-dimensional Lamb–Oseen vortex, and the study examines the influence of vortex strength and surface tension on the resulting flow. The wave dynamics are characterized as a function of the Reynolds and Weber numbers, and a phase diagram is presented in terms of (Re, We) to distinguish the different wave patterns. To ensure accurate reconstruction of the interface, the numerical methods used in this study feature a mass and momentum consistent advection method, high-order interpolation schemes, and a block-structured adaptive mesh refinement strategy. The study presents the characteristics of the air cavity entrained at the moment of wave impact for each wave pattern. Furthermore, the results provide insight into the nature of bubble entrainment within a vortex and reveals the bubble entrainment process via a breakup cascade. Meanwhile, it is also shown that the entrainment of bubble results in significant vortex distortion. Overall, this research contributes to enhance our understanding of wave dynamics and the intricate interaction between vortices and interfaces.

Investigation of the ultrasound-induced collapse of air bubbles near soft materials

A numerical investigation into the ultrasound-induced collapse of air bubbles near soft materials, utilizing a novel multi-material diffuse interface method (DIM) model with block-structured adaptive mesh refinement is presented. The present work expands from a previous five-equation DIM by incorporating Eulerian hyperelasticity. The model is applicable to any arbitrary number of interacting fluid and solid material. A single conservation law for the elastic stretch tensor enables tracking the deformations for all the solid materials. A series of benchmark cases are conducted, and the solution is found to be in excellent agreement against theoretical data. Subsequently, the ultrasound-induced bubble-tissue flow interactions are examined. The bubble radius was found to play a crucial role in dictating the stresses experienced by the tissue, underscoring its significance in medical applications. The results reveal that soft tissues primarily experience tensile forces during these interactions, suggesting potential tensile-driven injuries that may occur in relevant treatments. Moreover, regions of maximal tensile forces align with tissue elongation areas. It is documented that while early bubble dynamics remain relatively unaffected by changes in shear modulus, at later stages of the penetration processes and the deformation shapes, exhibit notable variations. Lastly, it is demonstrated that decreasing standoff distances enhances the interaction between bubbles and tissue, thereby increasing the stress levels in the tissue, although the behavior of the bubble dynamics remains largely unchanged.

An Adaptive Mesh Refinement–Rotated Lattice Boltzmann Flux Solver for Numerical Simulation of Two and Three-Dimensional Compressible Flows with Complex Shock Structures

An adaptive mesh refinement–rotated lattice Boltzmann flux solver (AMR-RLBFS) is presented to simulate two and three-dimensional compressible flows with complex shock structures. In the method, the RLBFS, which has a strong shock-capturing capability and can effectively eliminate the shock instability phenomenon, is applied to solve the flow filed by reconstructing the fluxes at each cell interface adaptively with the mesoscopic lattice Boltzmann model. To locally and dynamically improve the resolution of intricate shock structures and optimize the required computational resources, a block-structured adaptive mesh refinement (AMR) technique is introduced. The validity and effectiveness of the proposed method are confirmed through a range of two and three-dimensional numerical cases, including the shock tube problem, the four-wave Riemann problem, explosion within a rectangular box, and the vorticity induced by a shock. The results obtained using the AMR-RLBFS exhibit excellent agreement with published data and demonstrate high accuracy in capturing complex shock structures. The computational efficiency of the AMR-RLBFS can be also improved significantly compared to the RLBFS on uniform grids. Furthermore, the numerical outcomes underscore the capability of the AMR-RLBFS to eliminate shock instability effects while efficiently capturing a broader spectrum of small-scale vertical structures. These findings highlight the ability of AMR-RLBFS to improve the computational efficiency and capture intricate shock structures effectively, making it a valuable tool for studying a wide range of compressible flows from aerodynamics to astrophysics.

Numerical study of liquid jet and shock wave induced by two-bubble collapse in open field

The shock waves and high-speed jets induced by the violent collapse of bubbles can cause serious damage to nearby structures. To fully understand the shock wave and jet characteristics induced by bubble-bubble interaction, we developed a diffuse-interface method for the simulation of compressible two-phase flow, which combined interface compression technique with interface sharpening technique to keep the sharpness of the shock wave and phase interface. The thermodynamically consistent five-equation model was improved using interface compression technique to maintain a constant thickness of the phase interface. To further suppress numerical dissipation, the high-order scheme WENO (Weighted Essentially Non-Oscillatory) and interface sharpening function THINC (Tangent of Hyperbola for INterface Capturing) were used for the flux reconstruction at the grid boundary to ensure that the total variation at the cell boundary was minimized (Boundary Variation Diminishing, BVD), thus the spatial reconstruction scheme, named WENO-THINC-BVD, was developed. Moreover, we extended the present method to a block-structured adaptive mesh refinement framework to improve grid resolution and save computing resources. Numerical results of several benchmark tests fully demonstrated the advantages of the present method in simulating complex compressible flows containing shock-shock and shock-interface interactions. Based on the present high-fidelity numerical methods, we investigated the shock wave and liquid jet induced by the two-bubble interaction in an open space. A phase diagram of the liquid jet propagation direction as a function of the initial bubble pressure and the inter-bubble distance was summarized. Finally, according to the bubble-bubble interaction process, an empirical formula for the collapse strength and another for the attenuation of shock waves were developed.

A numerical investigation of deflagration propagation and transition to detonation in a microchannel with detailed chemistry: Effects of thermal boundary conditions and vitiation

We numerically investigate the premixed flame acceleration (FA) and the subsequent deflagration to detonation transition (DDT) of pure and vitiated fuel/oxidizer mixtures in a microchannel under two extreme wall thermal conditions—an adiabatic wall and a hot, preheated isothermal wall. The numerical simulations are conducted using AMReX-Combustion PeleC, an exascale compressible reacting flow solver that leverages load-balanced block-structured adaptive mesh refinement to enable high-fidelity direct numerical simulation. We perform these simulations for a hydrogen combustion system. While it is widely known that adiabatic walls strongly promote the occurrence of DDT via FA, such a mechanism of DDT is found to be strongly limited by the flame speeds of the unreacted mixture and hence is intrinsically tied to the mixture composition. We demonstrate that the addition of water (i.e., vitiation) to the unreacted mixture leads to a significant reduction in the flame speed, thereby slowing down the FA process and subsequent DDT. With isothermal preheated walls, the pure fuel cases preferentially propagate along the wall after an auto-ignition event, leading to the formation of a “secondary” finger-flame. This secondary front subsequently undergoes transverse expansion, following which deceleration of the flame is observed. The vitiated fuel cases also exhibit a similar behavior, nonetheless exhibit much longer time-scales of auto-ignition and propagation, in addition to stronger deceleration. In summary, this study presents one of the very few simulations in the FA and DDT literature that employ detailed chemical kinetics for both adiabatic and isothermal walls.

High-performance GPU computing of phase-field lattice Boltzmann simulations for dendrite growth with natural convection

The effect of natural convection on dendrite morphology is investigated through three-dimensional large-scale phase-field lattice Boltzmann simulations using a block-structured adaptive mesh refinement scheme with the mother-leaf method in a parallel-GPU environment. The simulations confirmed that downward buoyancy enhances the growth of the primary and secondary arms, and upward buoyancy delays the growth of those arms. In addition, the effect of natural convection on the solidification morphologies gradually decreased as the primary arm tip reached the top of the computational domain and finally stopped. Furthermore, in the longer simulation under purely isothermal diffusive conditions, detachment of the secondary arms owing to curvature-driven fragmentation was observed. A large-scale non-isothermal dendrite growth simulation was also conducted, wherein it was observed that the tip growth rate of the primary arm was delayed, and the secondary arm spacing was larger than that in the isothermal condition.

Phase-field lattice Boltzmann simulation of three-dimensional settling dendrite with natural convection during nonisothermal solidification of binary alloy

Understanding the motion and growth behaviors of equiaxed dendrites during solidification is important for predicting macrosegregation. In this study, we develop a phase-field lattice Boltzmann (PF-LB) simulation method for the settling and growth of an equiaxed dendrite during the nonisothermal solidification of a binary alloy. The PF-LB computations are accelerated by employing parallel computation using multiple graphic processing units (GPUs) and the octree block-structured adaptive mesh refinement method, which incorporates multiple mesh and time increment methods. By using the developed method, we can simulate the three-dimensional long-distance settling dendrite while considering the effects of latent heat release and natural convection. From the simulation results, we confirm that the natural convection due to the high solute concentration around a dendrite reduces the settling velocity. In addition, we observe that the temperature increase owing to latent heat release slows dendrite growth, which in turn slightly slows the settling velocity. From these results, we confirm that the effects of latent heat release and natural convection are not negligible in the quantitative evaluation of settling dendrites.

Parthenon—a performance portable block-structured adaptive mesh refinement framework

On the path to exascale the landscape of computer device architectures and corresponding programming models has become much more diverse. While various low-level performance portable programming models are available, support at the application level lacks behind. To address this issue, we present the performance portable block-structured adaptive mesh refinement (AMR) framework Parthenon, derived from the well-tested and widely used Athena++ astrophysical magnetohydrodynamics code, but generalized to serve as the foundation for a variety of downstream multi-physics codes. Parthenon adopts the Kokkos programming model, and provides various levels of abstractions from multidimensional variables, to packages defining and separating components, to launching of parallel compute kernels. Parthenon allocates all data in device memory to reduce data movement, supports the logical packing of variables and mesh blocks to reduce kernel launch overhead, and employs one-sided, asynchronous MPI calls to reduce communication overhead in multi-node simulations. Using a hydrodynamics miniapp, we demonstrate weak and strong scaling on various architectures including AMD and NVIDIA GPUs, Intel and AMD x86 CPUs, IBM Power9 CPUs, as well as Fujitsu A64FX CPUs. At the largest scale on Frontier (the first TOP500 exascale machine), the miniapp reaches a total of 1.7 × 1013 zone-cycles/s on 9216 nodes (73,728 logical GPUs) at [Formula: see text] weak scaling parallel efficiency (starting from a single node). In combination with being an open, collaborative project, this makes Parthenon an ideal framework to target exascale simulations in which the downstream developers can focus on their specific application rather than on the complexity of handling massively-parallel, device-accelerated AMR.

A consistent mass-momentum advection method for the simulation of large-density-ratio two-phase flows

In present study, a simple and robust method is proposed to calculate the mass and momentum flux in the simulation of large-density-ratio interfacial flows. An auxiliary continuity equation is introduced and solved in a consistent manner with the momentum equations. This method ensures the consistency of the mass and momentum transportation in both phases. The erroneous interfacial momentum transfer often arising in traditional two-phase flow simulations can be suppressed. Moreover, this method can be extended straightforwardly to work with any of the VOF (Volume of Fluid) - type interface capturing methods. Present algorithm is implemented on an in-house code with block-structured adaptive mesh refinement (BAMR) (Liu and Hu, 2018) technologies, and is highly applicable for the massively parallel computations. We demonstrate the numerical stability and accuracy through a series of typical two-phase flow tests with high density ratios. Finally, the simulation of atomization of liquid sheet is considered to further demonstrate the capability in simulating surface tension driven flows. Numerical results show the spurious air velocity and unphysical interface fragmentation are completely avoided. Present consistent method exhibits much better performance in suppressing the excessive kinetic energy introduced by the two-phases co-existing cells. Moreover, the method is simple for implementation and is robust for the violent two-phase flow problems, even with strong shear velocity around the interface.

A mass-conserving and volume-preserving lattice Boltzmann method with dynamic grid refinement for immiscible ternary flows

In this paper, a lattice Boltzmann model with dynamic grid refinement is proposed for immiscible ternary flows, which is capable of conserving the total mass and preserving the volume of each phase. The application of interpolation schemes in adaptive mesh refinement (AMR) techniques results in the violation of the total mass of the fluids system within the lattice Boltzmann method (LBM) framework. In the present model, a source term with two free parameters is introduced into the interface capturing equation, which can be determined by the mass conservation and the volume preservation properties. A piecewise constant function has been deliberately incorporated into the source term in order to avoid the appearance of an unphysical fluid at the interface of other two fluids. Based on a block-structured AMR method, the governing equations for phase-field variables and flow hydrodynamic properties are solved by the finite-difference multiple-relaxation-time (MRT) LBM. Simulations of several typical problems are performed in order to evaluate the accuracy and applicability of the proposed model. The numerical results demonstrate that the present model can conserve both mass and volume at the same time as well as reduce numerical dispersion in the bulk region.

An approximated volume of fluid method with the modified height function method in the simulation of surface tension driven flows

Surface tension in two-phase flow problems plays a dominant role in many micro-flow phenomena and has an important influence on the development of flow instability phenomena that contain free surfaces. In this study, the multi-moment finite volume method is extended for direct numerical simulation of two-phase flow problems. A constraint interpolation profile–CSL (semi-Lagrangian) scheme is used for discretization of the advection part in the momentum equation. A compact volume of fluid method–approximated piecewise linear calculation method without flux limiter is proposed for capturing the moving interface. For modeling the surface tension accurately, the logic in curvature estimation is redesigned based on the height function (HF) method. The isolated volumetric fractions that may reduce accuracy in HF integration are excluded, and the numerical solution shows that the accuracy in the curvature estimation is improved for a coarse mesh. The present method is implemented with a parallel block-structured adaptive mesh refinement (BAMR) strategy; thus, the computational cost can be reduced significantly. Numerical tests show that the present BAMR solver is capable of reproducing the theoretical predictions of capillary wave instability problems with high accuracy. The simulation of droplet collisions further demonstrates the accuracy of the surface tension model. Finally, we extend it to the liquid jet atomization. The wavy disturbance, film breakup, liquid filament pinch-off, and droplet generation are well reproduced. The droplet size distribution satisfies the experimental measurement and theoretical predictions power-law. BAMR shows a huge advantage in computational efficiency than the traditional Cartesian grid. The findings of this study can help for a better understanding of the micro-mechanism of surface tension driven flows.

Efficient Shock Capturing Based on Parallel Adaptive Mesh Refinement Framework

Based on AMReX, a software framework for massively parallel, block-structured adaptive mesh refinement (AMR) applications and in combination with finite difference weighted essential non-oscillatory (WENO) method, a numerical procedure is developed to provide universal discontinuity capturing capability for inviscid, compressible flow. Test cases of one-dimensional and two-dimensional flows containing shock waves, contact discontinuities, flow instabilities, and their interactions are considered to validate the high resolution of AMR for characteristic flow structure under unsteady conditions. The current elaborate AMR-based code is proven to have superior efficiency relative to global refinement via experiments including different initial mesh intervals, refinement ratios at all levels, and. Evidence in executions with multiple processes and domain division sizes indicates its high parallel scalability. Thus, the application built on AMReX is promising for simulations of phenomena undergoing rapid local variation such as compressible turbulence and multiphase interactions.

A diffuse interface method for solid-phase modeling of regression behavior in solid composite propellants

Solid composite propellants (SCPs) are ubiquitous in the field of propulsion. In order to design and control solid rocket motors, it is critical to understand and accurately predict SCP regression. Regression of the burn surface is a complex process resulting from thermo-chemical-mechanical interactions, often exhibiting extreme morphological changes and topological transitions. Diffuse interface methods, such as phase field (PF), are well-suited for modeling processes of this type, and offer some distinct numerical advantages over their sharp-interface counterparts. They also provide a convenient framework for incorporating multiple multiphysical dynamics. In this work, we present a phase-field method for modeling the regression of SCPs with varying species and geometry. We construct the model from a thermodynamic perspective, leaving the base formulation general. A diffuse-species-interface field is employed as a mechanism for capturing complex burn chemistry in a reduced-order fashion, making it possible to model regression from the solid phase only. The computational implementation, which uses block-structured adaptive mesh refinement and temporal substepping for increased performance, is briefly discussed. The model is then applied to four test cases: (i) pure AP monopropellant, (ii) AP PBAN sandwich, (iii) AP HTPB sandwich, and (iv) spherical AP particles packed in HTPB matrix in two and three dimensions. In all cases, reasonable quantitative agreement is observed, even when the model is applied predictively (i.e., no parameter adjustment), as in the case of (iv). The validation of the proposed PF model demonstrates its efficacy as a numerical design tool for future SCP investigation.

A moving embedded boundary approach for the compressible Navier-Stokes equations in a block-structured adaptive refinement framework

A computational technique has been developed to perform compressible flow simulations involving moving boundaries using an embedded boundary approach within the block-structured adaptive mesh refinement (SAMR) framework of AMReX [1,91,92]. We leverage the SAMR capability to obtain quantitatively accurate results whilst using robust, second-order finite volume schemes. A conservative, unsplit, cut-cell approach is utilized and a ghost-cell approach is developed for computing the flux on the moving, embedded boundary faces. A third-order least-squares formulation has been developed to compute the wall velocity gradients, and was found to significantly improve the performance of the solver in terms of the quantitative comparison of surface quantities such as the skin friction coefficient. Various test cases are performed to validate the method, and compared with analytical, experimental, and other numerical results in literature. Inviscid and viscous test cases are performed that span a wide regime of flow speeds – acoustic (harmonically pulsating sphere), smooth flows (expansion fan created by a receding piston) and flows with shocks (shock-cylinder interaction, shock-wedge interaction, pitching NACA 0012 airfoil and shock-cone interaction). A closed system with moving boundaries – an oscillating piston in a cylinder, showed that the percentage error in mass within the system decreases with refinement, demonstrating that the numerical scheme is conservative with grid refinement, but is not discretely conservative. Viscous test cases involve that of a horizontally moving cylinder at Re=40, an inline oscillating cylinder at Re=100, and a transversely oscillating cylinder at Re=185. The judicious use of adaptive mesh refinement with appropriate refinement criteria to capture the regions of interest leads to well-resolved flow features, and good quantitative comparison is observed with the results available in literature.

A consistent mass–momentum flux computation method for the simulation of plunging jet

In the present study, a robust and conservative numerical scheme is proposed to simulate the violent two-phase flows with high-density ratios. In this method, the mass conservation equation and the momentum equation are solved in a consistent manner. The tangent of hyperbola for interface capturing scheme is extended for the computation of the mass flux by which the sharpness and conservation property of density field is preserved. Compared with other recently proposed methods, no geometrical computation is involved in deriving the mass flux and the spurious velocity in the interfacial region can be completely avoided. To improve the computational efficiency, the present method is implemented on a parallel block-structured adaptive mesh refinement method with a staggered layout of variables. High-fidelity numerical simulation of plunging jet through the liquid surface is performed. A bubble detection algorithm is developed to track bubbles generated in air entrainment process. The evolution of the bubble cloud, air concentration, bubble-size, and bubble-velocity distributions are predicted and compared quantitatively with the experiment. Numerical results show the air entrainment and penetration depth are highly correlated with the upstream disturbance. The growing interfacial roughness of the jet yields more entrained air in the final stage of jet impingement. It is found that when the initial perturbation is introduced, the overall size of the equivalent bubble radius will expand, and the penetration depth of the bubble cloud will decrease, while a larger volume of air is entrained.

Quokka: a code for two-moment AMR radiation hydrodynamics on GPUs

ABSTRACT We present quokka, a new subcycling-in-time, block-structured adaptive mesh refinement (AMR) radiation hydrodynamics (RHD) code optimized for graphics processing units (GPUs). quokka solves the equations of HD with the piecewise parabolic method (PPM) in a method-of-lines formulation, and handles radiative transfer via the variable Eddington tensor (VET) radiation moment equations with a local closure. We use the amrex library to handle the AM management. In order to maximize GPU performance, we combine explicit-in-time evolution of the radiation moment equations with the reduced speed-of-light approximation. We show results for a wide range of test problems for HD, radiation, and coupled RHD. On uniform grids in 3D on a single GPU, our code achieves >250 million hydrodynamic updates per second and almost 40 million radiation hydrodynamic updates per second. For RHD problems on uniform grids in 3D, our code scales from 4 to 256 GPUs with an efficiency of 76 per cent. The code is publicly released under an open-source license on GitHub.

A hybrid adaptive multiresolution approach for the efficient simulation of reactive flows

Computational studies that use block-structured adaptive mesh refinement (AMR) approaches suffer from unnecessarily high mesh resolution in regions adjacent to important solution features. This deficiency limits the performance of AMR codes. In this work a novel hybrid adaptive multiresolution (HAMR) approach to AMR-based calculations is introduced to address this issue. The multiresolution (MR) smoothness indicators are used to identify regions of smoothness on the mesh where the computational cost of individual physics solvers may be decreased by replacing direct calculations with interpolation. We suggest an approach to balance the errors due to the adaptive discretization and the interpolation of physics quantities such that the overall accuracy of the HAMR solution is consistent with that of the MR-driven AMR solution. The performance of the HAMR scheme is evaluated for a range of test problems, from pure hydrodynamics to turbulent combustion.

Numerical investigation of flow structure and air entrainment of breaking bow wave generated by a rectangular plate

Breaking bow waves entrain massive gas that generate ambient noise and produce spray and bubbly wake with whitecap. This study aims to give a quantitative description of the flow structures and bubble formation during the breaking process. We consider the breaking bow waves induced by a surface-piercing flat plate and perform simulations based on an in-house code. We employ a conservative coupled level-set and volume of fluid method to capture violent variation of the liquid–gas interface. A robust immersed boundary method is adopted to model the motion of the plate. To resolve very small flow structures associated with the wave breaking process with the available computational resources, a block-structured adaptive mesh refinement strategy is used. It is found that the predicted wave characteristics, such as wave height, wave crest location, and wave profile, are consistent with the experiment. A wide range of flow phenomena, including the thin liquid sheet, jet overturning, and splash-ups are well reproduced by the present simulation. In addition, we implement a bubble-droplet detection program to track single bubbles, and the characteristics of bubble cloud (entrained air volume, spatial distribution, and penetration depth) can be analyzed quantitatively. Three typical bubble creation mechanisms for the air entrainment process of the breaking bow wave are reported, and ensemble-averaged statistics of the bubble size distribution are presented. We also quantify the evolution of the bubble distribution and discuss the power-law scaling during the bow wave breaking process.

Tree cutting approach for domain partitioning on forest-of-octrees-based block-structured static adaptive mesh refinement with lattice Boltzmann method

The aerodynamics simulation code based on the lattice Boltzmann method (LBM) using forest-of-octrees-based block-structured adaptive mesh refinement (AMR) with temporary-fixed refinement was implemented, and its performance was evaluated on GPU-based supercomputers. Although the Space-Filling-Curve-based (SFC) domain partitioning algorithm for the octree-based AMR has been widely used on conventional CPU-based supercomputers, accelerated computation on GPU-based supercomputers revealed a bottleneck due to costly halo data communication. Our new tree cutting approach adopts a hybrid domain partitioning with the coarse structured block decomposition and the SFC partitioning in each block. This hybrid approach improved the locality and the topology of the partitioned sub-domains and reduced the amount of the halo communication to one-third of the original SFC approach. In the strong scaling test, the code achieved maximum ×1.82 speedup at the performance of 2207 MLUPS (mega-lattice update per second) on 128 GPUs (NVIDIA® Tesla® V100). In the weak scaling test, the code achieved 9620 MLUPS at 128 GPUs with 4.473 billion grid points, while keeping the parallel efficiency of 93.4% from 8 to 128 GPUs.

Block structured adaptive mesh refinement and strong form elasticity approach to phase field fracture with applications to delamination, crack branching and crack deflection

Fracture is a ubiquitous phenomenon in most composite engineering structures, and is often the responsible mechanism for catastrophic failure. Over the past several decades, many approaches have emerged to model and predict crack failure. The phase field method for fracture uses a surrogate damage field to model crack propagation, eliminating the arduous need for explicit crack meshing. In this work a novel numerical framework is proposed for implementing hybrid phase field fracture in heterogeneous materials. The proposed method is based on the “reflux-free” method for solving, in strong form, the equations of linear elasticity on a block-structured adaptive mesh refinement (BSAMR) mesh. The use of BSAMR enables highly efficient and scalable regridding, facilitates the use of temporal subcycling for explicit time integration, and allows for ultra-high refinement at crack boundaries with minimal computational cost. The method is applied to a variety of simple heterogeneous structures: laminates, wavy interfaces, and circular inclusions. In each case a non-dimensionalized parameter study is performed to identify regions of behavior, varying both the geometry of the problem and the relative fracture energy release rate. In the laminate and wavy interface cases, regions of delamination and fracture correspond to simple analytical predictions. For the circular inclusions, the modulus ratio of the inclusion is varied as well as the delamination energy release rate and the problem geometry. In this case, a wide variety of behaviors was observed, including deflection, splitting, delamination, and pure fracture.