Dual-horizon peridynamics: A stable solution to varying horizons

Huilong Renc; Xiaoying Zhuangd; Timon Rabczuk

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Dual-horizon peridynamics: A stable solution to varying horizons

Huilong Renc, Xiaoying Zhuangd, Timon Rabczuk

arXiv:1703.05910v1Mar 17, 2017physics.comp-ph

TL;DR

Traditional peridynamics struggles with spurious reflections and computational inefficiency when horizon sizes or point resolutions vary. The paper develops dual-horizon peridynamics, proves its dual property and conservation balances, and evaluates variable horizons, multiple materials, and irregular point distributions. The reported examples show minimal spurious reflection, stable crack patterns, and handling of multiple materials with limited modification.

Problem
Traditional PD requires constant horizons, making variable-resolution discretizations costly and causing spurious wave reflections when horizon radii differ.
Method
DH-PD introduces dual-horizons, proves their general dual property, and formulates direct and reaction forces for variable horizons and multiple materials.
Results
DH-PD handles spurious reflections, multiple materials, and irregular particle arrangements, producing stable crack patterns that agree with experiments.
Takeaways & Limitations
DH-PD supports variable horizons and spatially irregular discretizations while fulfilling balance of momentum and angular momentum.
Takeaways & Limitations
The angular-momentum proof uses a bounded, Riemann-integrable force function on Hilbert function space.

Abstract

from arXiv · show

In this paper, we present a dual-horizon peridynamics formulation which allows for simulations with dual-horizon with minimal spurious wave reflection. We prove the general dual property for dual-horizon peridynamics, based on which the balance of momentum and angular momentum in PD are naturally satisfied. We also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics. For selected benchmark problems, we show that DH-PD is less sensitive to the spatial than the original PD formulation.

1. Introduction

Peridynamics models fracture through nonlocal interactions and bond breaking, but traditional formulations require constant horizons, limiting efficient use of varying spatial resolutions. The paper introduces dual-horizon peridynamics to address spurious reflections while supporting variable horizons, reduced cost, conservation laws, random point distributions, and multiple materials.

Peridynamics naturally captures complex fracture patterns by breaking bonds between material points, without explicitly describing crack topology.
PD formulations use integral equations in which each material point interacts with neighboring points inside a horizon.
Constant horizons are traditionally required because varying horizon sizes can produce spurious wave reflections and ghost forces.
DH-PD defines dual-horizons from neighboring points’ horizons, enabling variable horizons while reducing computational cost and retaining constant-horizon PD as a special case.
The paper proves a general dual property and uses it to re-establish balance of linear and angular momentum, while also examining random point distributions and multiple materials.

2. Dual-horizon peridynamics

Dual-horizon peridynamics separates direct-force horizons from reaction-force dual-horizons so interactions remain consistent when point resolutions and horizon radii vary. Its equation of motion combines direct and reaction forces, with each bond force calculated once and conventional constant-horizon PD recovered as a special case.

Implementation: Each bond force in DH-PD is calculated once, with the corresponding reaction force automatically included through Newton’s third law.
Peridynamic formulations: DH-PD applies directly to bond-based, ordinary state-based, and non-ordinary state-based peridynamics.
The shortcomings of constant horizons: Constant-horizon PD can require dramatically more neighbors under nonuniform point sizes, increasing computational cost without comparable accuracy gains.
The shortcomings of constant horizons: Variable horizon radii cause spurious reflections in conventional PD because a single-horizon formulation poorly accounts for interactions between points with different radii.
Horizon and dual-horizon: In DH-PD, a horizon contains points forming direct bonds, while a dual-horizon collects points whose horizons contain the current point.
Bond force density: DH-PD independently breaks bonds and dual-bonds because different horizon sizes can imply different critical stretches.
Equation of motion: The DH-PD equation of motion sums inertia, body, direct, and reaction forces, and remains valid near boundaries.

3. The dual property of dual-horizon

The dual property converts double integration over dual-horizons into integration over horizons with bond arguments swapped, providing the basis for the formulation’s conservation proofs.

3. The dual property of dual-horizon: The dual property states that dual-horizon integration of F(x, x′) equals horizon integration of F(x′, x).The same relation is constructed in continuum and discretized forms by regrouping bond terms.
3. The dual property of dual-horizon: The proof discretizes the domain into N Voronoi tessellations with centers xi and associated volumes ΔVi.Each material point is assigned a horizon and dual-horizon, and bond-dependent terms are summed over these regions.
3. The dual property of dual-horizon: For any bond-dependent expression F(i, j), terms indexed through a point’s dual-horizon can be regrouped according to the corresponding horizons.The regrouping relies on interpreting each weighted term in the dual-horizon of one point and the horizon of the other.
3. The dual property of dual-horizon: Swapping the indices i and j yields the discrete dual identity, which becomes the continuum relation after replacing indices with x and x′.The argument proceeds through index exchange and relabeling before stating the integral form.
3. The dual property of dual-horizon: The dual property provides a shorter proof that dual-horizon peridynamics satisfies balance of linear and angular momentum.The paper presents this proof as a consequence of the dual-horizon identity.

Balance of linear momentum

The internal forces in dual-horizon peridynamics satisfy the balance of linear momentum for any bounded body.

Balance of linear momentum: The internal forces satisfy the balance of linear momentum for any bounded body Ω.The paper states this balance in equation form and proves it from the preceding dual-horizon relation.
Balance of linear momentum: Linear momentum is conserved because the relevant balance equation is satisfied directly from Eq. (24).The proof identifies Eq. (26) as apparently satisfied based on Eq. (24).

Balance of angular momentum

The paper establishes angular-momentum balance in dual-horizon peridynamics under the stated constitutive assumptions, with conservation following from the force structure.

Balance of angular momentum: Angular-momentum balance is required for any bounded body Ω.The requirement is introduced before the constitutive proposition and proof.
Balance of angular momentum: The constitutive proposition assumes a bounded, Riemann-integrable vector-state function with other variables collected in Λ.Under the stated condition, angular momentum holds for any deformation and given constitutive model.
Balance of angular momentum: Angular momentum is conserved because the expression obtained in step 4 vanishes for BB-PD, OSB-PD, and NOSB-PD.The proof uses the preceding force relation and states that the zero-expression result applies to all three formulations.
Balance of angular momentum: For BB-PD and OS-PD, internal forces are parallel to the current bond vector, while the dual-horizon is needed only in the equation-of-motion terms.The paper notes that angular-momentum conservation depends on the horizon in this argument.

4. Wave propagation in 1D homogeneous bar

A one-dimensional bar with strongly different spatial resolutions tests wave transmission across variable-horizon interfaces. DH-PD limits interface effects, whereas traditional PD produces spurious reflection and waves.

4. Wave propagation in 1D homogeneous bar: The bar has length L = 1 m, with a central fine region discretized at Δx1 = 4.19×10^-3 m and outer regions at Δx2 = 10Δx1.Both ends are free, and the horizon is selected as δi = 3.015Δxi.
4. Wave propagation in 1D homogeneous bar: DH-PD and traditional PD are compared as Case I and Case II using the same variable-resolution bar setup.Wave profiles are reported at different times for the two formulations.
4. Wave propagation in 1D homogeneous bar: The DH-PD horizon interfaces have limited influence on wave profiles, while traditional PD produces a spurious wave.The comparison is shown in the wave-profile figures for the two cases.
4. Wave propagation in 1D homogeneous bar: The reflected wave in DH-PD is smaller than 3.17% of the incident-wave magnitude.The incident wave passes the interface, based on displacement measurements at x = 0.32 m and x = 0.34 m.
4. Wave propagation in 1D homogeneous bar: Traditional PD generates ghost forces near the interface, causing spurious waves and strong reflection of the fine-region wave.The fine-region wave is difficult to transmit because its horizon is much smaller than the coarse-region horizon.

5. Multiple materials

DH-PD addresses multiple-material and variable-spacing effects by treating bond and dual-bond interactions separately, producing solutions closer to analytical fields and stable heterogeneous crack patterns.

Multiple-material formulation: DH-PD computes heterogeneous interactions using force states and can combine material parameters from both sides of a bond.For BB-PD, alternate arithmetic or geometric averages of microelastic moduli or critical stretches can be used.
Multiple-material formulation: OSB-PD and NOSB-PD calculate force states similarly for homogeneous and heterogeneous materials, without distinguishing the two cases.
1D bar tensile test: Case I produces force discontinuities and deviations from the analytical solution at interfaces with different material properties or point spacing.The one-sided microelastic modulus leads to spurious equilibrium displacement and force fields.
1D bar tensile test: DH-PD solutions for Case II agree well with analytical displacement and force solutions, with only small interface variations attributed to nonlocality.The observed variation is distinguished from ghost force because horizon and dual-horizon forces remain balanced.
Conclusion: DH-PD is presented as having potential to effectively handle multiple materials in the simple 1D example.
Multiple-material crack patterns: In heterogeneous plates, the DH-PD damage rule breaks bonds and dual-bonds separately according to the parameters of the point exerting each force.The simulations show increased crack branching and crack propagation through inclusions.

6. Simulation of the Kalthoff–Winkler experiment

The Kalthoff–Winkler benchmark evaluates DH-PD under irregular material-point distributions and compares simulated crack paths and speeds with experimental behavior and other methods.

Benchmark setup: The benchmark models a steel 18Ni1900 specimen under impact loading, with the experiment reporting brittle fracture and an approximately 70° crack angle at 32 m/s.The simulations use an initial velocity of 22 m/s and material parameters E = 190 GPa, ρ = 7800 kg/m3, ν = 0.25, and G0 = 6.9 × 10^4 J/m2.
Material-point distributions: Irregular material points are generated by converting finite-element meshes into points with volumes or masses allocated to element nodes.The tested particle-size ratios are 3.1 in 2D and 4.4 in 3D; Cases I and III use constant-horizon PD.
Crack patterns: DH-PD produces crack patterns that agree well with the experiment, whereas traditional constant-horizon bond-based PD differs in both 2D and 3D.The reported crack angle from DH-PD is better than the CH-PD result relative to the approximately 70° reference for lower-speed impacts.
Crack propagation speed: The simulated crack speeds peak at 1669, 1658, 1699, and 1664 m/s for Cases I–IV, respectively, all within 75% of Rayleigh speed.Crack propagation begins at 20 µs.

7. Conclusions

The paper establishes dual-horizon peridynamics as a formulation for variable horizons, multiple materials, and irregular point distributions. Its analysis and numerical examples show conservation properties, reduced computational cost, and more stable crack predictions than traditional constant-horizon peridynamics.

The dual property of dual-horizon peridynamics was proved, supporting balance of momentum and angular momentum.These conservation properties are re-established from the dual property.
DH-PD handles spurious wave reflection, multiple-material problems, and crack stability on irregular material-point arrangements.The numerical examples address ghost forces, material interfaces, and crack patterns under random particle distributions.
Bond-based DH-PD produces stable crack patterns agreeing with experiments despite irregular point distributions, unlike traditional bond-based PD.Traditional bond-based PD crack patterns are affected by the irregular material-point distribution.
DH-PD reduces neighbor counts and force-summation work relative to constant-horizon PD in inhomogeneous discretizations.The force density is calculated once in DH-PD instead of twice in traditional PD.
DH-PD requires only minimal modification to handle multiple materials.This advantage is stated relative to traditional peridynamics.

A.1. Volume correction

The volume-correction procedure improves force integration when material points partly overlap a horizon. In DH-PD, it accounts for the intersection between irregular material-point domains and point-specific horizons.

Volume correction accounts for material points that partly fall inside a horizon to improve force-summation accuracy.The correction is introduced because finite-sized points may only partially belong to a horizon.
In DH-PD, each material point has a unique horizon, so volume ratios between neighboring points can become large.This differs from the constant-horizon setting and motivates a geometry-based correction.
A partially included neighboring point contributes its intersection domain to the reaction-force integration.The example adds the green intersection region to the integration over H_i.
The correction uses the intersection area or volume of two circles or spheres with radii R and r separated by distance d.The geometric construction is illustrated by the circle-to-circle intersection figure and its associated formulas.