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Boundary-Level-Constrained Refinement for Suppressing Trimming-Induced High-Frequency Outliers in Explicit Isogeometric Analysis

Christoph Hollweck, Lukas Leidinger, Stefan Hartmann, Marcus Wagner, Roland Wüchner

arXiv:2609.10161v1math.NAcs.NA

TL;DR

The paper examines how trimming and local refinement together affect the upper spectral limit, which governs the critical time step. It introduces BLCR to constrain refinement near trimming boundaries; in the investigated configurations, BLCR shifts control of the maximum eigenfrequency to the untrimmed interior and increases the admissible time step.

  • Problem

    The paper asks how trimming and local refinement jointly affect the upper spectral limit, since reduced-support basis functions near trimming boundaries may remain spectrally critical.

    Source: §1 · p. 4, §5 · p. 39

  • Method

    BLCR constrains the refinement level of basis functions whose support intersects the trimming boundary relative to the refined interior.

    Source: §1 · p. 5, §5 · p. 39

  • Results

    Trimmed basis functions can still increase the maximum eigenfrequency without arbitrarily small cut cells, while investigated BLCR configurations yield a lower maximum eigenfrequency and larger critical time step than uniformly refined trimmed patches with the same interior resolution.

    Source: §5 · p. 39

  • Takeaways & Limitations

    BLCR is a targeted time-step strategy for interior refinement when the trimmed boundary is not a primary approximation region, trading boundary resolution for time-step efficiency.

    Source: §1 · p. 5, §4.4 · p. 35

    BLCR can alter lower frequencies and reduce boundary approximation quality, so it warrants care for low-frequency-dominated problems or when accurate dynamics near trimmed boundaries are required.

    Source: §3.1.3 · p. 19, §4.4 · p. 35

Abstract

from arXiv · show

In explicit dynamics, the critical time step is governed by the maximum eigenfrequency of the semi-discrete system. In Isogeometric Analysis (IGA), open knot vectors introduce a characteristic spectral boundary effect that can lead to high-frequency outliers and restrict the admissible time step. In the row-sum-lumped setting considered here, extending the computational patch and trimming away the exterior boundary functions mitigates this time-step penalty, but does not necessarily prevent boundary-adjacent basis functions from governing the maximum eigenfrequency. We show that reduced-support basis functions adjacent to the trimming boundary can remain critical spectral contributors even for knot-exact trimming, i.e., in the absence of small cut cells. Hence, small cut cells are not required for trimming-induced time-step penalties. Knot-exact and arbitrary trimming represent different degrees of support reduction within the same underlying mechanism. To control this effect, we propose the Boundary-Level-Constrained Refinement (BLCR) strategy, a local refinement constraint for LR- and THB-splines. BLCR constrains the refinement level of basis functions whose support intersects the trimming boundary relative to that of the refined interior. In all configurations investigated in this work, this constraint ensures that the maximum eigenfrequency is governed by untrimmed interior basis functions rather than by refined trimmed functions. Consequently, the trimming-induced high-frequency outliers governing the critical time step are suppressed, yielding a larger admissible time step than globally refined trimmed B-splines with the same interior resolution.[...]

Highlights

The paper identifies reduced-support basis functions at trimming boundaries as remaining contributors to high-frequency outliers, even after exterior open-knot boundary functions are removed. It proposes BLCR to constrain refinement near those boundaries and reports larger admissible time steps while maintaining close agreement in investigated response quantities.

1. Introduction

  • Reduced-support basis functions adjacent to trimming boundaries can contribute to the maximum eigenfrequency even with knot-exact trimming and no arbitrarily small cut fraction.Trimming changes stiffness and row-sum-lumped mass contributions non-proportionally, so small cut cells are not required for the effect.Source: Boundary-Level-Constrained Refinement… · p. 2, §1 · p. 4

Aim and Outline of the Paper

  • Computable eigenfrequency bounds explain how reduced support at trimming boundaries contributes to the upper spectral limit for knot-exact and arbitrary trimming.The study focuses on the maximum eigenfrequency that governs the critical explicit time step, not general spectral-accuracy limitations of mass lumping.Source: Boundary-Level-Constrained Refinement… · pp. 2–3, Aim and Outline of the Paper · p. 5
  • BLCR constrains the refinement level of basis functions whose support intersects the trimming boundary relative to the refined interior, and applies to LR- and THB-splines.It targets the upper spectral limit while preserving the diagonal mass formulation without additional stabilization.Source: Boundary-Level-Constrained Refinement… · p. 2
  • Critical time steps increase by up to 42% compared with globally refined trimmed patches.The paper also reports larger stable time steps in an LS-DYNA cross-bowl forming simulation, with close agreement in the investigated response quantities.Source: Boundary-Level-Constrained Refinement… · p. 2

1. Introduction

  • Explicit time steps are limited by the semi-discrete system’s maximum eigenfrequency, so high-frequency outliers can increase computational cost.The paper situates its investigation in row-sum-lumped trimmed LR- and THB-spline discretizations.Source: §1 · p. 4, Aim and Outline of the Paper · p. 5
  • Removing exterior open-knot boundary functions does not eliminate trimming-related contributions to the maximum eigenfrequency.The active portion of a boundary-adjacent basis function’s support affects its stiffness-to-mass ratio.Source: §1 · p. 4

2. Theoretical Framework

The theoretical framework introduces B-spline representations and refinement concepts used to construct locally refined discretizations. It covers univariate and tensor-product bases, then describes LR-splines through local knot insertion.

2. Theoretical Framework

  • The framework covers spline formulations, trimming, B-Rep models, explicit stability limits, and the numerical models used in later analyses.It presents B-splines as the foundation for LR- and THB-splines.Source: §2 · p. 5

Univariate B-splines

  • B-splines are piecewise polynomials of degree p, with continuity determined by knot multiplicity, and form a partition of unity.The univariate basis is defined using the Cox–de Boor recursion.Source: Univariate B-splines · pp. 5–6

Tensor-product B-splines

  • Tensor-product basis functions combine knot vectors in two parametric directions, producing rectangular elements with polynomial bases of degree (p, q).The study assumes equal degrees in both directions.Source: Tensor-product B-splines · p. 6

2.2. LR-splines

  • LR-splines use local knot insertion to refine the basis.Inserting a knot splits an affected parent basis function into weighted children and creates a nested mesh hierarchy.Source: §2.2 · p. 6, Univariate Refinement · p. 7

Bivariate Refinement

Bivariate refinement constructs local spline meshes and bases using LR meshlines or hierarchical THB spaces; trimming represents complex surfaces by active parameter-space regions bounded by loops. The section also describes structured refinement and how THB truncation manages active functions.

Bivariate Refinement

  • LR refinement inserts axis-aligned meshlines that split intersected rectangular elements, producing a locally refined partition of the parametric domain.Bivariate basis functions are defined by local knot vectors and carry weights.Source: Bivariate Refinement · p. 7

Structured Refinement Strategy

  • Structured refinement bisects marked supports in both parametric directions, assigns child elements and new basis functions the next level, and preserves constant element aspect ratios.A basis function’s level is the minimum level of the elements in its support.Source: Structured Refinement Strategy · p. 8

2.3. THB-splines

  • THB-splines use nested spaces and hierarchical refinement regions; truncation removes contributions from child functions active on finer levels, preserving partition of unity.The stated construction uses explicit parent–child relationships, and its discrete space typically has fewer active degrees of freedom while preserving classical THB approximation properties.Source: §2.3 · pp. 8–10
  • THB refinement can produce non-convex supports and require control-point updates for newly activated functions; basis functions may be truncated multiple times.The THB and LR illustrations use the same hierarchical mesh for direct comparison.Source: §2.3 · p. 9

2.4. Trimmed Surfaces

  • Trimming defines closed parameter-space loops that partition a rectangular surface domain into active and void regions, with loop orientation determining the active side.The active surface maps the active parameter-space region into physical space.Source: §2.4 · p. 10

Linear elastodynamics

The section sets out a linear elastodynamic initial-boundary-value problem in one or two dimensions, with displacement, density, body force, and stress as its principal quantities.

Linear elastodynamics

  • The model is a linear elastodynamic initial-boundary-value problem on a domain in one or two dimensions over a finite time interval.The displacement field is defined over the domain and time interval.Source: Linear elastodynamics · p. 10
  • The strong form uses density, body-force density, and Cauchy stress, with the constitutive relation specified for linear elasticity.Source: Linear elastodynamics · pp. 10–11
  • In one dimension, the model reduces to a bar formulation using cross-sectional area and Young’s modulus, with displacement and traction boundary parts distinguished.Source: Linear elastodynamics · p. 11

Weak Formulation

The weak formulation follows by testing the strong form with a function in the stated Sobolev space and integrating by parts.

Weak Formulation

  • The weak form is obtained by multiplying the strong equation by a test function and integrating by parts.Source: Weak Formulation · p. 11
  • The stated test-function space consists of square-integrable functions whose first derivatives are also square-integrable.Source: Weak Formulation · p. 11
  • It seeks displacement in the space V for every time in the interval and every test function in V.Source: Weak Formulation · p. 11

Spline Discretization

The discretization uses locally refined LR- or THB-spline bases, retaining functions whose supports intersect the active trimmed domain. It forms a matrix system from the weak form and uses a free-free spectral setup to isolate discretization, trimming, and refinement effects.

Spline Discretization

  • After trimming, the active index set retains basis functions whose supports intersect the active parametric domain.Source: Spline Discretization · p. 11
  • LR basis functions are tensor products on their supports, while THB functions restrict to tensor-product polynomials on each element despite truncation.Source: Spline Discretization · p. 12
  • Insertion of the spline expansion into the weak form yields a matrix system whose entries are specified by the discretization.The displacement coefficients carry the time dependence, while the basis functions describe the spatial approximation.Source: Spline Discretization · p. 12

2.6. Numerical Model Setup

  • The spectral investigation uses an undamped free-free configuration without essential boundary conditions or external loads to isolate discretization, trimming, and refinement effects.The numerical models are one-dimensional bars and two-dimensional membranes; integrals are formulated parametrically and mapped using the geometric Jacobian.Source: Spline Discretization · p. 12, §2.6 · p. 12

1D Bar Model

The one-dimensional bar model uses an open-knot discretization and row-sum lumping to define its mass matrix. Numerical eigenfrequencies are validated against the analytical free-free bar solution.

1D Bar Model

  • The bar is discretized into n_e elements with equidistant open knot vectors and linear elastic material parameters E = 1, ρ = 1, and A_c = 1.A diagonal mass matrix is obtained by standard row-sum lumping.Source: 1D Bar Model · p. 12
  • The stiffness matrix entries are specified for the bar model.The supplied passage does not provide the entries themselves.Source: 1D Bar Model · p. 13
  • Numerical eigenfrequencies are compared with the analytical solution for a free-free bar of length L.The supplied passages do not provide the complete analytical expression.Source: 1D Bar Model · p. 13

2D Membrane Model

The membrane model uses row-sum-lumped masses and plane-stress stiffness on a square domain. The section also describes how the maximum system eigenfrequency sets the critical time step and how it can be conservatively estimated.

2D Membrane Model

  • The square membrane has thickness t = 1 and Poisson ratio ν = 0.3, with identical lumped mass assigned to each control point in both displacement directions.Parametric and physical coordinates coincide in the numerical experiments, so |det J| = 1.Source: 2D Membrane Model · p. 13
  • The membrane stiffness uses a plane-stress constitutive matrix and a strain-displacement matrix in Voigt notation.The element formulation accounts for the number of basis functions supported on each element.Source: 2D Membrane Model · p. 13

2.7. Stability of Explicit Time Integration

  • For explicit central-difference integration of an undamped system, the stable time step must satisfy the CFL condition set by the maximum system eigenfrequency.The maximum frequency is obtained from the generalized eigenvalue problem involving the mass and stiffness matrices.Source: §2.7 · p. 13
  • Because computing the maximum system eigenfrequency at every time step is costly, conservative nodal or element estimates are used, with characteristic element length often serving as a heuristic.For trimmed IGA meshes, the characteristic length of a trimmed element is not obvious; Jacobian-based alternatives account for polynomial degree and continuity.Source: §2.7 · p. 14
  • An industrial safety factor, typically f_s = 0.9, accounts for nonlinearities when setting the stable time step.The characteristic element length is a computationally inexpensive alternative to spectral analysis.Source: §2.7 · p. 14
  • The modified Gershgorin bound provides a more precise maximum-frequency bound when mass is non-uniform, as it can be after trimming.The classical Gershgorin circle theorem also supplies a conservative upper bound for the maximum system eigenfrequency.Source: §2.7 · p. 14

3. Numerical Investigation of Trimming-Induced High-Frequency Outliers

The investigations show that reduced-support basis functions next to a trimming boundary can remain critical spectral contributors, including with knot-exact trimming. BLCR constrains their refinement relative to the interior; in the investigated trimmed configurations, a one-level difference keeps the upper spectral limit governed by interior functions.

3.1.1. Refinement and Trimming Setups

  • Reduced-support basis functions adjacent to the trimming boundary can govern the maximum eigenfrequency even after exterior open-knot functions are removed, and refining them with the interior can make them limiting contributors.Knot-exact and arbitrary trimming are treated as different degrees of support reduction within the same mechanism.Source: §3 · p. 15, §3.1.1 · pp. 17–18
  • For p = 2, trimming the standard bar configuration reduces ω_max from 1.16 to 0.75, while global refinement scales the largest eigenfrequency by a factor of two.In the globally refined knot-exact trimmed case, identical element eigenfrequencies do not reveal the assembled patch's elevated maximum associated with reduced support.Source: §3.1.1 · p. 15
  • For p = 3, locally refined untrimmed configurations have ω_max = 1.57 for LR-splines and 1.53 for THB-splines, versus 1.08 and 1.02, respectively, after trimming.In the investigated p = 3 boundary-fitted case, a one-level refinement difference is insufficient; after knot-exact trimming, the additional relative coarsening is sufficient.Source: §3.1.1 · pp. 16–17

3.1.2. Boundary-Level-Constrained Refinement (BLCR)

  • BLCR confines the smallest parametric length scales to the interior by requiring boundary-intersecting basis functions to remain below the interior's maximum refinement level.For a single refinement step, only interior functions are refined; with multiple steps, trimmed functions may be refined if they remain below the maximal interior level.Source: §3.1.2 · p. 18
  • Keeping trimmed basis functions one refinement level below the maximally refined interior suffices to make untrimmed interior functions govern the upper spectral limit in all trimmed configurations investigated.The authors caution that this is not a universal guarantee across polynomial degrees, trimming configurations, or refinement hierarchies.Source: §3.1.2 · p. 18

3.1.3. Eigenvalue Spectrum under Refinement and Trimming

  • Knot-exact trimming can produce lower-spectrum deviations without partially cut elements or vanishingly small cut fractions, while BLCR can alter individual lower frequencies.The authors advise care for low-frequency-dominated problems and applications where accuracy near the trimming boundary matters.Source: §3.1.3 · p. 19

3.1.4. Refinement from left to right

  • When refinement first reaches trimmed basis functions with non-zero active support, ω_max jumps to 1.49 for p = 2 and 1.32 for p = 3.For refinement width w < δ, only functions supported entirely in the void region are refined and ω_max remains constant.Source: §3.1.4 · p. 20

Observations:

Refining trimmed basis functions can make them govern the maximum eigenfrequency, while keeping refinement away from the trimming boundary avoids that effect in the investigated configurations. Across the 1D and 2D studies, BLCR separates refined interior functions from trimmed boundary functions, and the complex-geometry cases show larger critical time steps than global refinement.

Observations:

  • Refining a trimmed basis function triggers a jump in ωmax; refining additional trimmed functions has only weak further influence.Even one refined trimmed function incurs most of the time-step penalty.Source: Observations · p. 20

3.1.5. Symmetric Refinement from Center to Boundary

  • With refinement restricted to the interior, ωmax stays below the case with refined trimmed functions; at w = 88, outliers return (ωmax = 1.49 for p = 2; 1.32 for p = 3).The investigated configurations motivate keeping the refined region separated from the trimming boundary.Source: §3.1.5 · p. 21

3.1.6. Continuous Trimming from Boundary to Center

  • During continuous trimming, ωmax rises once refined functions are cut; for p = 3, the initial decrease and plateau show a one-level boundary/interior difference alone is insufficient while the most restrictive boundary functions remain active.For p = 2, a one-level difference suffices in the investigated configuration; once the coarse trimmed functions no longer govern, the refined untrimmed interior controls ωmax.Source: §3.1.6 · p. 22

3.1.7. Spectral Bounds for Reduced-Support Basis Functions

  • For the investigated BLCR configuration at p = 3, ωmax lies in [0.816, 1.155], versus [0.963, 1.556] for uniformly refined trimmed B-splines; coarse trimmed functions are not limiting.The one-level rule is supported for the investigated configurations, but is not established universally across geometries, degrees, or refinement patterns.Source: §3.1.7 · pp. 24, 26

3.2. Investigations in 2D

  • The bivariate studies use expanded open-knot patches with void boundary rows and columns, then test square, rotated-square, and complex trimmed geometries.The complex-geometry BLCR cases restrict refinement to active, untrimmed basis functions.Source: §3.2 · pp. 26–27, §3.2.5 · p. 30

3.2.1–3.2.2. Symmetric Refinement from Center to Boundary

  • In center-to-boundary 2D refinement, ωmax rises when refined trimmed functions become critical; rotating the square produces separate jumps as edge and then corner functions are trimmed.The corner-related jump occurs at w = 52 in the investigated rotated-square configuration.Source: §3.2.1 · p. 27, §3.2.2 · p. 28

3.2.3–3.2.4. Continuous Trimming from Boundary to Center

  • In 2D continuous trimming, ωmax stays flat until refined functions are cut, then rises; the square also rises strongly for δ > 19, while rotation smooths transitions.For the rotated configuration, the relevant distinction is whether reduced-support functions belong to the refined or coarse level, not whether trimming follows element edges.Source: §3.2.3 · pp. 28–29, §3.2.4 · p. 29

3.2.5. Complex Trimmed Geometry

  • BLCR increased the critical time step by approximately 42% for p = 2 and 38% for p = 3 versus global refinement on the complex trimmed geometry.BLCR restricted refinement to active, untrimmed functions.Source: §3.2.5 · p. 30, Table 1 · p. 32

4. Industrial Application

The industrial cross-bowl forming benchmark tests BLCR in an explicit LS-DYNA simulation and compares its time-step efficiency and response with globally refined trimmed B-splines. The section also discusses when BLCR’s coarser boundary resolution is acceptable and how trimming-boundary refinement relates to cross-talk.

4.1. Model setup for the Cross Bowl

  • The benchmark uses an explicit LS-DYNA Reissner–Mindlin shell simulation of a curved sheet-metal blank drawn into a die, with contact, plasticity, and large deformations.The tools are modeled as rigid surfaces, and Coulomb friction is used at the contact interfaces.Source: §4 · p. 32, §4.1 · p. 32

4.2. Preprocessing

  • BLCR preprocessing requires the initial geometry to be represented one refinement level coarser than the target, then refines only untrimmed interior basis functions.The trimmed boundary basis functions are deliberately left on the coarse level; the coarser representation must still capture the geometry sufficiently accurately.Source: §4.2 · p. 33

4.3. Results

  • BLCR increases the admissible time step by approximately 32% for p = 2 and 28% for p = 3, with close agreement in the investigated total-energy and thickness results.The comparison is with globally refined trimmed B-splines; total-energy deviations are at most 1.6% for p = 2 and 1.0% for p = 3.Source: Table 2 · p. 34, §4.3 · pp. 34–35

4.4. Applicability and limitations

  • BLCR keeps boundary-adjacent basis functions coarse, trading lower edge resolution for a smaller maximum eigenfrequency and larger admissible time step.The reduced boundary accuracy may be acceptable in sheet forming because blank edges are typically scrap; in crash models, accurate boundary stresses, strains, or damage may instead require boundary refinement and mass scaling.Source: §4.4 · pp. 35–36
  • In the investigated configurations, the time-step reduction after one refinement step is about 31% with BLCR versus about 50% under uniform global refinement.Combining BLCR with adaptive refinement may further reduce the number of elements needed for localized phenomena, while limiting the time-step impact.Source: §4.4 · p. 36

4.5. Cross-talk effects

  • Cross-talk can arise when small trimming features disconnect basis-function supports, creating non-physical coupling between separated domain regions.The section distinguishes complete support separation (Type 1) from spurious tractions from partially trimmed supports (Type 2); boundary refinement or mass scaling may address these effects, but can introduce spectral outliers.Source: §4.5 · p. 37

5. Conclusion and Outlook

The paper identifies reduced-support basis functions at trimming boundaries as a remaining source of high maximum eigenfrequencies, including under knot-exact trimming. It introduces BLCR to keep those functions coarser than the refined interior, improving critical time steps while retaining application-dependent trade-offs in boundary resolution and spectral accuracy.

Conclusion

  • Reduced-support basis functions adjacent to trimming boundaries can contribute to the maximum eigenfrequency even after open-knot boundary elements are removed and without arbitrarily small cut cells.The mechanism is the non-proportional change in stiffness and row-sum-lumped mass contributions caused by support reduction.Source: §5 · p. 39
  • The investigated Rayleigh and modified Gershgorin bounds explain why a one-level refinement difference suffices in the tested p = 2 and p = 3 cases, but do not establish a universal one-level criterion.The bound separation is configuration dependent.Source: §5 · p. 39
  • In the investigated configurations, BLCR shifts the upper spectral limit from trimmed basis functions to the refined untrimmed interior, yielding a larger critical time step than uniformly refined trimmed patches at the same interior resolution.The strategy preserves the row-sum-lumped diagonal mass matrix and does not modify total mass.Source: §5 · p. 39
  • Both LR- and THB-splines are suitable for BLCR, and their critical time steps are effectively identical in the investigated two-dimensional configurations.Their refinement mechanisms produce minor differences in one dimension, but those differences become negligible in the tested two-dimensional cases.Source: §5 · p. 39

Conclusion and Outlook

  • The cross-bowl benchmark supports BLCR for the considered sheet-metal forming problems, but does not establish general spectral accuracy in nonlinear dynamics.Coarser boundary resolution is an application-dependent trade-off, and changes to lower-spectrum frequencies remain relevant to assess for the excitation and quantities of interest.Source: §5 · pp. 39–40
  • Local mass scaling is an alternative when local refinement is unavailable or unwanted, but it alters total mass and inertia, and suitable scaling factors remain an open problem.Practical upper-spectrum approaches should also preserve diagonal mass structure and avoid requiring global mass or stiffness assembly.Source: §5 · p. 40

Appendix A. Asymptotic Behavior of the Maximum Eigenfrequency under Mesh Refinement

The appendix gives scaling arguments for the maximum discrete eigenfrequency under uniform mesh refinement and relates local stiffness-to-mass ratios to system-level bounds. For interior B-splines, the characteristic frequency scales inversely with element size in one and multiple dimensions.

Appendix A. Asymptotic Behavior of the Maximum Eigenfrequency under Mesh Refinement

  • For interior B-splines, the stiffness-to-mass ratio scales as h^-2 under uniform refinement, giving a characteristic frequency scaling of h^-1.The appendix makes explicit the scaling relations used in the main text rather than deriving a new estimate.Source: Appendix A · p. 40

Appendix A.1. Bounds on the Maximum Eigenfrequency

  • A lower bound for the maximum eigenfrequency is given by the maximum of the diagonal stiffness-to-mass ratios.The appendix frames the maximum eigenfrequency through the Rayleigh quotient of Kq = λMq.Source: Appendix A.1 · p. 40

Appendix A.2. Scaling Analysis in One Dimension

  • In one dimension, an interior B-spline’s lumped mass entry scales as h and its stiffness entry as h^-1, so its local frequency is inversely proportional to element length.The basis magnitude remains O(1) while its derivative scales as O(h^-1) as support shrinks.Source: Appendix A.2 · p. 41

Appendix A.3. Generalization to Multiple Dimensions

  • In d dimensions, the diagonal mass and stiffness entries scale as h^d and h^(d-2), respectively, preserving the h^-2 ratio and h^-1 frequency scaling.The mass scaling follows from support measure, while the stiffness scaling follows from squared gradients integrated over that support.Source: Appendix A · p. 40, Appendix A.3 · pp. 41–42

Appendix A.4. Scaling of the Maximum Eigenfrequency

For fixed polynomial degree, the maximum eigenvalue has matching lower and upper mesh-size estimates for untrimmed interior basis functions. This scaling argument is limited: trimming can change active supports and stiffness-to-mass contributions non-proportionally.

Appendix A.4. Scaling of the Maximum Eigenfrequency

  • An interior basis function’s diagonal Rayleigh quotient scales as h^-2, so the largest eigenvalue is bounded below by a quantity proportional to h^-2.The maximum over all degrees of freedom cannot be smaller than that interior quotient.Source: Appendix A.4 · p. 42
  • The upper scaling uses the standard inverse inequality for fixed polynomial degree on shape-regular discretizations.The inequality completes the scaling argument for the maximum eigenvalue.Source: Appendix A.4 · pp. 42–43
  • The lower and upper estimates have the same mesh-size dependence, establishing the familiar asymptotic scaling of the maximum eigenvalue for untrimmed interior basis functions.The lower bound follows from an interior basis function’s diagonal Rayleigh quotient, which scales as h^-2; the upper estimate uses an inverse inequality.Source: Appendix A.4 · pp. 42–43
  • The scaling argument applies to maximally smooth, untrimmed interior basis functions on uniform or shape-regular meshes with fixed polynomial degree, but does not directly extend to arbitrarily trimmed basis functions.Trimming changes active support and may alter stiffness and row-sum-lumped mass contributions non-proportionally, so frequency bounds can depend on the trimming configuration.Source: Appendix A.4 · p. 43
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