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An Unfitted Hybrid High-Order Method for the Elastodynamics Problem with Imperfect Interface

Peiqi Huang, Erik Burman

arXiv:2610.01504v1math.NAcs.NA

TL;DR

Unresolved interfaces, extreme compliancy values, and arbitrarily small cut cells pose challenges for elastic-wave discretization. The paper develops an unfitted HHO method for linear-slip interfaces and proves robust stability and error estimates. It also identifies a limitation in uniform locking behavior for the stabilization analyzed.

  • Problem

    Elastic-wave discretization across unresolved interfaces must represent displacement jumps, handle wide compliancy ranges and small cut cells, and preserve discrete energy.

    Source: §1 · p. 2

  • Method

    An arbitrary-order unfitted HHO method combines a local symmetric strain reconstruction incorporating the interface condition with stabilization weighted by the regularized interface stiffness.

    Source: §1 · p. 3

  • Results

    The method has energy-error order h^(k+1) and L2-error order h^(k+2), with constants independent of compliancy parameters and interface cuts.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The formulation recovers the bonded and decoupled traction-free interface limits within one scheme, with both limits confirmed numerically.

    Source: §1 · p. 3

    The analysis does not establish locking-free behavior for this stabilization, and experiments show error growth proportional to λ^(1/2).

    Source: §5.2 · p. 16

Abstract

from arXiv · show

We design and analyse an unfitted hybrid high-order (HHO) method for the elastic wave equation in a medium made of two components separated by an imperfect interface of linear slip type, across which the traction is continuous and the displacement jump is proportional to the traction through a compliancy tensor $\bK=\alpha\bI+(\beta-\alpha)\bn\otimes\bn$. The mesh is not fitted to the interface: the discrete unknowns are doubled in the cut cells, the small cuts are cured by a cell agglomeration procedure, and no unknown is attached to the interface. The two specific ingredients of the method are a local symmetric strain reconstruction in each cut subcell, which incorporates the interface condition through the regularised interface stiffness $\bS_h=(h_T\delta^{-1}\bI+\bK)^{-1}$ in the spirit of Hansbo and Hansbo {\em{A finite element method for the simulation of strong and weak discontinuities in solid mechanics.}} {Comput. Methods Appl. Mech. Engrg.}, 193, 2004, and an interface stabilisation built from the same matrix. For the space semi-discrete problem we prove that the discrete bilinear form is coercive and continuous, and we derive an energy-error estimate of order $h^{k+1}$ and an $L^2$-error estimate of order $h^{k+2}$, with constants independent of the compliancy parameters and of how the interface cuts the mesh. The scheme is combined either with the Newmark scheme, which conserves a discrete energy exactly, or with singly diagonally implicit Runge--Kutta schemes of order up to four. Numerical experiments in two dimensions confirm the predicted convergence rates for $k\in\{1,2,3\}$, the robustness with respect to the compliancy over sixteen orders of magnitude, and illustrate the propagation of elastic waves across an unresolved slipping interface.

1 Introduction

The paper develops an arbitrary-order unfitted HHO method for elastic waves across a linear-slip interface, combining a cut-cell strain reconstruction with interface stabilization. Its analysis establishes robust stability and error estimates, while the introduction frames the method against challenges from unresolved interfaces, extreme compliancy, and small cuts.

1 Introduction

  • Linear-slip conditions model imperfect bonding by continuous traction and a displacement jump proportional to traction; the compliancy tensor interpolates between perfect bonding and decoupled subdomains.For isotropic interfaces, the tangential and normal compliancies are α and β; in the large-compliancy limit each subdomain has a traction-free boundary.Source: §1 · p. 2
  • The discretization must represent displacement jumps inside unresolved cut cells, remain stable across wide compliancy ranges, control arbitrarily small subcells, and preserve discrete energy.These are the four difficulties identified for numerical approximation of the wave equation.Source: §1 · p. 2
  • Prior work treated unfitted elasticity, HHO discretization, and wave propagation separately, while this paper combines these ingredients for elastodynamics with a linear-slip interface.The cited related work includes unfitted elasticity using a regularized interface stiffness and earlier unfitted HHO methods for elliptic interface problems.Source: §1 · pp. 2–3
  • The method combines a local symmetric strain reconstruction that incorporates the interface condition with an interface stabilization weighted by the regularized interface stiffness Sh.The reconstruction is defined on each cut subcell, and the stabilization penalizes the jump of cell unknowns across the interface.Source: §1 · p. 3
  • The semi-discrete method has an energy-error estimate of order h^(k+1) and an L2-error estimate of order h^(k+2), with constants independent of compliancy and interface cuts.The introduction also reports coercivity and continuity of the discrete bilinear form; time integration uses energy-conserving Newmark or singly diagonally implicit Runge–Kutta schemes of order up to four.Source: §1 · p. 3

2 An elastodynamics problem with an imperfect interface

The section sets the geometry and function spaces for elastodynamics on two subdomains separated by a smooth internal interface. It states the displacement problem, its interface conditions, and conventions for handling singular compliancy tensors.

2 An elastodynamics problem with an imperfect interface

  • The domain is connected, convex, and polygonal or polyhedral in dimension two or three, with a smooth C2 interface dividing it into two subdomains.The interface is the shared boundary of the subdomains, and the paper defines jumps as the trace from Ω1 minus the trace from Ω2.Source: §2 · p. 3
  • The weak-solution setting uses piecewise H1 displacement fields with zero outer-boundary trace, with time regularity specified for displacement and its derivatives.The section defines the relevant Sobolev spaces and the time interval J = (0,T).Source: §2 · pp. 3, 5
  • The elastodynamics problem has piecewise material parameters, prescribed initial data and forcing, and interface conditions prescribing the displacement jump and traction jump.The displacement jump is coupled to traction through K, while the stated data include gD and gN.Source: §2 · p. 4
  • When K is singular, terms involving K^-1 use its pseudoinverse and the trial and test spaces are restricted to fields whose jump lies in the range of K.The perfectly bonded case K = 0 corresponds to a prescribed displacement jump; the same discretization covers these cases without a case distinction.Source: §2 · p. 5

3 The unfitted HHO method

The method combines an unfitted mesh and doubled cut-cell unknowns with a local strain reconstruction and interface stabilization designed for linear-slip interfaces. Its semi-discrete HHO scheme uses these local operators to approximate the elastic wave equation.

3.1 Unfitted meshes

  • The mesh is unfitted to Γ; agglomerating cells with small cuts enforces the cut-cell assumptions without propagating beyond nearest neighbours.The geometric agglomeration is independent of the discretisation.Source: §3.1 · p. 6

3.2 Discrete unknowns and the regularised interface stiffness

  • Cut cells carry degree k+1 cell unknowns and degree k face unknowns on each side, with no unknown attached to the interface.The regularised stiffness Sh is defined using K and hT/δ and remains well defined for all compliancy parameters.Source: §3.2 · pp. 6–7

3.3 Local reconstruction and stabilisation

  • A local symmetric strain reconstruction embeds the interface condition on the softer side, while Sh-weighted stabilization penalises the cell-unknown jump across Γ.The same formulation recovers weak displacement continuity in the bonded regime and decoupling in the high-compliancy limit.Source: §3.3 · pp. 7–8

3.4 The space semi-discrete scheme

  • The space semi-discrete scheme combines cell and face unknowns to approximate the second-order elastic wave equation while weakly matching them across subdomain faces.Its interface terms and stabilization weakly enforce the first interface condition.Source: §3.4 · pp. 8–9

4 Preliminaries

The preliminaries establish parameter-independent conventions and the matrix, discrete-inequality, stability, and approximation tools used in the analysis. In particular, the local interpolation construction is chosen to avoid approximation dependence on how Γ cuts a cell.

4 Preliminaries

  • The analysis uses constants independent of the Lamé coefficients, compliancy tensor, mesh size, and how the interface cuts the mesh.The constants may depend on the polynomial degree and mesh parameters.Source: §4 · p. 9

4.1 The regularised interface stiffness

  • The regularised interface stiffness Sh is symmetric and commutes with K, while ShK is positive semi-definite.These properties hold for the defined regularisation Sh = (hT δ^-1 I + K)^-1.Source: §4.1 · p. 9

4.2 Discrete and trace inequalities

  • Discrete trace, Poincaré, and Korn inequalities control polynomial traces, cell values, and gradients on cut subcells under the stated cut-cell assumption.The Poincaré and Korn bounds require the respective orthogonality conditions.Source: §4.2 · p. 10

4.3 Local energy norm and stability of the local operators

  • The local energy norm weights face contributions differently: its μ-weighted term uses the full trace-cell difference, while its λ-weighted term uses only the stabilization quantity.This distinction keeps the stability constants independent of the Lamé coefficients.Source: §4.3 · p. 11

4.4 Interpolation and approximation

  • The local interpolation projects onto a larger ball T† rather than a cut subcell, making its approximation properties independent of how Γ cuts the cell.The interpolation operator uses extensions of the subdomain fields and face projections.Source: §4.4 · p. 12

5 Stability and error analysis

The analysis establishes stability and well-posedness for the space semi-discrete method, then derives energy- and L2-error estimates with optimal orders. The estimates are uniform in interface compliancy and cut geometry, but the analysis does not claim locking-free behavior.

5.1 Stability of the discrete bilinear form

  • The discrete bilinear form is symmetric, coercive, and continuous with stability constants independent of mesh size, Lamé coefficients, and interface compliancy; its norm gives well-posedness.The stability result is Theorem 5.1, and the norm property yields a well-posed space semi-discrete problem.Source: §5.1 · p. 14

5.2 Consistency

  • Interface-condition terms cancel in the consistency argument after integration by parts, using traction continuity and the two interface conditions.The resulting consistency bound includes a material factor that deteriorates as κ_i = λ_i/μ_i grows; the method is not claimed to be locking-free.Source: §5.2 · pp. 15–16
  • The standard stabilisation is not locking-free: its consistency bound retains an intrinsic factor growing with λ_i, and the analysis makes no locking-free claim.The fitted-case remedy uses an anisotropic stabilisation and an H(div)-conforming interpolant, which is unavailable on cut subcells with curved boundaries.Source: §5.2 · p. 16

5.3 Energy-error estimate

  • The energy-error estimate has order h^(k+1), optimal for the polynomial degrees used, with constants independent of compliancy and interface cut geometry.The estimate depends on material parameters through explicit factors, and the bounds degenerate as κ_i tends to infinity.Source: §5.3 · p. 16, §5.4 · p. 19

5.4 An elliptic projection and the L2-error estimate

  • The L2-error estimate has order h^(k+2), optimal for the polynomial degrees used, with constants independent of compliancy and interface cut geometry.A Ritz-type elliptic projection and duality argument are used to obtain the higher-order estimate.Source: §5.4 · pp. 17–19

6 Time discretisation

The semi-discrete system is realised algebraically with a block-diagonal mass matrix and a symmetric positive-definite stiffness matrix. Newmark advances hybrid displacement, velocity, and acceleration unknowns at each time node.

6.1 Algebraic setting and the Newmark scheme

  • The algebraic system pairs a block-diagonal mass matrix with a symmetric positive-definite stiffness matrix associated with the discrete bilinear form.The mass matrix corresponds to the density-weighted inner product.Source: §6.1 · p. 20
  • The Newmark scheme, parameterized by β and γ, is second-order accurate in time and implicit if β > 0.Source: §6.1 · p. 20
  • At each time node, the scheme approximates displacement, velocity, and acceleration as hybrid unknowns.The initial displacement and acceleration satisfy equations tested separately on cell and face components.Source: §6.1 · p. 20

3. Corrector: ˆun+1

The Newmark choice conserves the discrete energy under the stated unforced conditions, while the paper also considers higher-order SDIRK time stepping. SDIRK adds controlled dissipation of poorly resolved high frequencies and reuses a condensed matrix factorisation.

3. Corrector: ˆun+1

  • In the absence of external forcing and interface data, the discrete energy is exactly conserved, uniformly in mesh size, polynomial degree, and compliancy.Source: §3 · p. 20

6.2 Singly diagonally implicit Runge–Kutta schemes

  • The considered SDIRK(s, s + 1) methods have s stages and order s + 1, for s ∈ {1, 2, 3}, reaching order four.Only the time discretisation changes; the space discretisation and its analysis remain unchanged.Source: §6.2 · p. 21
  • Each SDIRK stage has Newmark's algebraic structure, and its condensed matrix can be factorised once for all stages and time steps.The cost per time step is s times that of a Newmark step.Source: §3 · p. 20, §6.2 · p. 21
  • Unlike conservative Newmark, SDIRK dissipates discrete energy without forcing, concentrating the loss on poorly resolved high-frequency components.The authors identify this dissipation as the mechanism behind greater accuracy than Newmark at a given time step.Source: §6.2 · pp. 21–22

7 Numerical experiments

The two-dimensional experiments assess convergence, compliancy and cut-geometry robustness, and wave propagation. In the stationary unfitted-slip test, the predicted energy- and L2-error rates are observed for k = 1, 2, 3.

7 Numerical experiments

  • The experiments cover error-rate, compliancy, and cut-geometry robustness, plus wave propagation across unresolved bonded and linear-slip interfaces.The two-dimensional test cases progress from a manufactured stationary problem to time-dependent and wave-propagation configurations.Source: §7 · p. 22

7.1 Implementation

  • The implementation uses uniform Cartesian meshes and exact cut geometry for a straight oblique interface, isolating the unfitted discretisation from geometric error.Small cuts are cured by merging a cut cell with a face neighbour; the merge procedure does not propagate beyond one neighbouring cell.Source: §7.1 · pp. 22–23

7.2 Test case 1: stationary problem with an unfitted slip interface

  • The stationary test uses an oblique interface crossing the square, with material contrast and default compliancy α = β = 5 · 10^-2.The manufactured solution satisfies the interface conditions for all nonnegative α and β.Source: §7.2 · p. 23
  • The stationary test observes energy-error rates k + 1 and L2-error rates k + 2 for k ∈ {1, 2, 3}.The errors are reported in Table 1 and plotted against h in Figure 3.Source: Table 1 · p. 24, §7.2 · p. 24, Figure 3 · p. 25

7.3 Robustness with respect to the compliancy

The experiments test robustness across compliancy regimes, small cuts, and the quasi-incompressible limit. They find largely parameter-independent errors, improved conditioning through agglomeration, and a trade-off between stabilisation weights in the quasi-incompressible case.

7.3 Robustness with respect to the compliancy

  • Across sixteen orders of magnitude in compliancy, relative errors vary by at most 50%; at α = β = 10^4 and 10^12 they agree to five significant digits, indicating numerical decoupling.The sweep also includes cases where one compliancy parameter vanishes and covers both Nitsche-like and compliancy-dominated regimes.Source: §7.3 · p. 25, Table 2 · p. 25

7.4 Small cuts and cell agglomeration

  • Without agglomeration, the condition number grows like ξ^-1, reaching 2·10^5 for k = 1 and 7·10^5 for k = 2 at ξ = 10^-4; agglomeration makes it uniform in ξ.Errors remain almost unaffected over the tested range with a sparse direct solver, but the ill-conditioning is problematic for iterative solvers and can affect accuracy at smaller ξ or higher degrees.Source: §7.4 · p. 26, Table 3 · p. 26

7.5 The quasi-incompressible limit

  • With stabilisation (18), quasi-incompressible errors grow like λ^1/2 while observed rates remain optimal; the anisotropic weight makes the L2-error uniform in λ over four orders of magnitude.The energy error still grows like λ^1/2 with the anisotropic weight, and proving L2-uniformity would require an H(div)-conforming interpolant unavailable on cut subcells.Source: §7.5 · p. 26, Table 4 · p. 27

7.6 Test case 1 in the time domain

The time-domain tests recover the predicted spatial rates and examine elastic-wave propagation across an unresolved interface. The finest tested configuration closely matches the reference, while SDIRK(3, 4) reaches space-error accuracy at a larger time step than Newmark.

7.6 Test case 1 in the time domain

  • The predicted spatial rates k + 1 and k + 2 are recovered with Δt = 1/2000, and the observed temporal rates approach Newmark’s second order.The time-step comparison uses k = 2, n = 8 and a Δt = 1/3200 reference solution.Source: §7.6 · p. 27, Table 6 · p. 27, Table 5 · p. 27

7.7 Test case 2: wave propagation across a perfectly bonded unresolved interface

  • At n = 64, the error on the most demanding velocity component falls from 16% for k = 1 to 2.5% for k = 2 and 0.09% for k = 3; all four components are within 0.1% of the reference.The interface is not aligned with mesh faces, and every cut cell carries doubled unknowns.Source: §7.7 · p. 28, Table 7 · p. 29
  • At Δt = 1/160, SDIRK(3, 4) has reached the space-error level, whereas Newmark errors deteriorate as the time step grows; one SDIRK step costs three Newmark steps.At Δt = 1/640 the schemes agree within half a percentage point, and the reported net saving is a factor of 4/3.Source: §7.7 · pp. 28–29, Table 8 · p. 30
  • The three tested SDIRK schemes exhibit their expected temporal orders 2, 3, and 4 once the time step resolves the discrete operator’s fastest modes.Source: §7.7 · p. 29, Table 9 · p. 31
  • Across the unresolved perfectly bonded interface, transmitted and reflected wave fronts cross without visible spurious reflection; Newmark conserves discrete energy to machine precision while SDIRK(3, 4) dissipates a small amount.The SDIRK(3, 4) energy loss is about 2·10^-5 of the initial energy at Δt = 1/640 and decreases by roughly a factor of 30 with each halving of Δt.Source: §7.7 · p. 29, Figure 6 · p. 30

7.8 Test case 3: wave propagation across a linear slip interface

For the linear-slip wave test, increasing compliancy increases interface slip, reduces transmission, and increases reflection. The method retains close agreement with the reference solution for a tested compliant interface.

7.8 Test case 3: wave propagation across a linear slip interface

  • As compliancy increases, the transmitted amplitude into Ω1 decreases and the reflected amplitude into Ω2 increases; at α = β = 5·10^-2 the transmitted front has almost disappeared.The recovered discrete slip along Γ also grows with compliancy.Source: §7.8 · p. 31, Figure 7 · p. 31, Figure 8 · p. 32
  • At α = β = 2·10^-2, the finest configuration reproduces the reference solution within 0.15% on all four sensor components.The comparison uses the k = 4, n = 64 reference solution.Source: §7.8 · p. 31, Table 10 · p. 32

8 Conclusions

The paper presents an unfitted HHO method for elastodynamics with linear-slip interfaces, with stability and error estimates uniform in compliancy and interface cuts. Numerical experiments confirm its convergence, robustness, and ability to propagate waves across an unresolved interface.

8 Conclusions

  • The discrete bilinear form is coercive and continuous, with constants independent of compliancy and interface cuts; errors converge as h^(k+1) in energy and h^(k+2) in L2.These estimates apply to the space semi-discrete problem.Source: §8 · p. 33
  • The method doubles unknowns in cut cells, combines local symmetric strain reconstruction with interface-stiffness-weighted stabilisation, and uses agglomeration to cure small cuts without interface unknowns.Its reconstruction incorporates the interface condition through I − ShK.Source: §8 · p. 33
  • Experiments confirm the predicted rates for k in {1, 2, 3}, robustness across sixteen orders of magnitude in compliancy, and wave propagation across an unresolved compliant interface.They also report the necessity and effectiveness of agglomeration for small cuts, including degenerate cases α = 0 or β = 0.Source: §8 · p. 33
  • Newmark time integration conserves discrete energy exactly, while singly diagonally implicit Runge–Kutta schemes up to order four dissipate controlled energy and tolerate larger time steps at the same accuracy.The larger time steps are several times larger.Source: §8 · p. 33
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