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A Local Stabilized Isogeometric Formulation with Discontinuous Pressure for Mesh-Conforming and Immersed Analysis of Nearly Incompressible Elasticity

Ashkan Ali Madadi, Thomas J. R. Hughes

arXiv:2610.36349v1cs.CEcs.NAmath.NA

TL;DR

Displacement-based formulations can lock near incompressibility, while mixed methods require compatible displacement-pressure spaces and high-order Galerkin discontinuous-pressure IGA demands increasing grid coarsening with polynomial degree. This paper develops a VMS-stabilized mixed isogeometric formulation that retains local pressure condensation and extends it to immersed analysis. Its studies report that a factor-of-two pressure coarsening suffices for all considered polynomial orders and that immersed treatments preserve the intended mixed stability properties.

  • Problem

    Mixed displacement-pressure methods avoid relying solely on displacement approximations near incompressibility, but high-order Galerkin IGA with discontinuous pressure requires increasingly strong pressure-grid coarsening as polynomial degree rises.

    Source: §1 · p. 2

  • Method

    The paper uses VMS residual stabilization to control nonconstant elementwise pressure modes while retaining discontinuous pressure, local condensation, and an immersed extension.

    Source: §1 · p. 5

  • Results

    A pressure grid coarsened by a factor of two suffices for all polynomial orders considered, and immersed discrete inf-sup studies show preservation of intended displacement-pressure stability.

    Source: §1 · pp. 5–6

  • Takeaways & Limitations

    The framework combines high-order approximation, residual stabilization, local pressure elimination, and immersed geometric flexibility for nearly incompressible elasticity.

    Source: §5 · p. 81

    The stated sufficient stability condition relies on a single patch, aligned nested tensor-product spaces, quasi-uniform knot vectors, and a geometry map with bounded derivatives and Jacobian bounded away from zero.

    Source: §2.4 · p. 16

Abstract

from arXiv · show

Nearly incompressible materials arise in soft tissues, elastomers, gels, and other soft solids. Yet, numerical modeling remains challenging because displacement-based formulations can exhibit volumetric locking near incompressibility. Mixed displacement-pressure formulations address this difficulty but require compatible approximation spaces satisfying the inf-sup condition. In mixed formulations, discontinuous pressure spaces are attractive since pressure degrees of freedom can be condensed locally. In high-order isogeometric analysis, however, their Galerkin stability requires pressure-grid coarsening that increases with polynomial degree. Here, we develop a variational multiscale (VMS) stabilized mixed isogeometric formulation that retains this locality while substantially relaxing the restriction. The key observation is that VMS stabilization controls nonconstant elementwise pressure modes and is unconditionally stable with respect to its parameter, provided the elementwise constant pressure component satisfies the inf-sup condition. Consequently, for maximally continuous displacement splines, a pressure grid coarsened by a factor of two satisfies the constant-mode inf-sup criterion independently of polynomial degree. Local condensation then yields a displacement-only system, facilitating use in existing displacement solvers. The formulation is further extended to immersed isogeometric analysis with extended B-splines, pressure-cell aggregation, subdivision-based cut-cell quadrature, and weak enforcement of essential boundary conditions. Discrete inf-sup studies and convergence analyses in the Kirsch and Cook's membrane benchmarks demonstrate stability and show that the immersed treatment preserves mixed stability properties. The proposed framework therefore provides a local, high-order, and robust approach for nearly incompressible problems in mesh-conforming and immersed settings.

1 Introduction

The paper develops a locally condensable stabilized mixed formulation that relaxes the degree-dependent pressure-grid coarsening required by Galerkin discontinuous-pressure IGA. It extends the approach to immersed analysis and reports that its immersed treatments preserve the intended mixed stability properties.

1 Introduction

  • Highly continuous spline spaces make familiar degree-based finite-element pairing rules unreliable, so stability depends on degree, continuity, and relative mesh resolution together.A pressure mesh coarsened by a factor of two is described as a successful alternative for certain spline pairs.Source: §1 · p. 2
  • Existing local discontinuous-pressure Galerkin schemes require progressively greater pressure-grid coarsening as polynomial degree increases, potentially limiting pressure accuracy.The paper positions its stabilization as a way to retain local condensation while relaxing this restriction.Source: §1 · pp. 3, 5
  • VMS stabilization is unconditionally stable for a positive parameter, but pressure-gradient stabilization controls nonconstant modes, not elementwise constants.The constant component must instead satisfy an inf-sup condition or be controlled with interelement pressure-jump terms; this work adopts the inf-sup approach.Source: §1 · pp. 4–5
  • A fixed factor-of-two pressure-grid coarsening suffices for all polynomial orders considered, while discontinuous pressure unknowns remain locally condensable.VMS stabilization controls nonconstant elementwise pressure modes; the elementwise constant mode is controlled through inf-sup compatibility.Source: §1 · pp. 3–5
  • The formulation extends to immersed IGA using extended B-splines, pressure-cell aggregation, subdivision-based cut-cell quadrature, and weak Nitsche enforcement of essential boundary conditions.These treatments address small cut supports, poorly supported pressure degrees of freedom, irregular cut-cell integration, and nonconforming boundaries.Source: §1 · pp. 4–5
  • The immersed discrete inf-sup studies show that extended B-splines and pressure-cell aggregation preserve the intended stability of displacement-pressure pairs.The studies include inf-sup control of the elementwise constant pressure mode.Source: §1 · pp. 5–6

2 Theory

The theory focuses on the formulation-specific ingredients for stabilized mixed isogeometric analysis. It sets the mixed problem in the shared mathematical structure of incompressible flow and nearly incompressible elasticity.

2 Theory

  • The theory develops the Stokes-type mixed structure used for nearly incompressible elasticity, where pressure is an auxiliary volumetric variable rather than flow pressure.The algebraic and stability structure is shared, although the variables have different physical interpretations in the two settings.Source: §2 · p. 7
  • The section addresses the saddle-point formulation, local pressure elimination, inf-sup-stable spaces, and combining stabilization and immersed cut-cell treatments with discontinuous pressure.Source: §2 · p. 7

2.1 Stokes-type mixed problem

The section casts nearly incompressible elasticity in a Stokes-type displacement-pressure mixed framework. It distinguishes pressure’s volumetric role in elasticity from its role as a constraint multiplier in incompressible flow.

2.1 Stokes-type mixed problem

  • The mixed weak formulation is obtained by taking the first variation of the saddle-point functional with admissible displacement and pressure test functions.Source: §2.1 · p. 8
  • The strong form is useful for residual-based stabilization because its momentum operator has the same differential structure as the Stokes operator.Source: §2.1 · pp. 8–9
  • In the incompressible limit, pressure enforces the zero-divergence constraint and its stability must come from displacement-pressure coupling.For finite compressibility, the volumetric term directly controls pressure; this contribution vanishes as λ tends to infinity.Source: §2.1 · p. 9
  • The nearly incompressible elasticity problem has the same saddle-point structure as incompressible flow, although the physical meanings of the variables differ.In flow, pressure acts as a Lagrange multiplier; in elasticity, it is an auxiliary volumetric variable.Source: §2.1 · pp. 9–10
  • In the solid-mechanics setting, u is displacement, µ is shear modulus, and p is an auxiliary pressure-like variable associated with volumetric response.The pressure is associated with λ-weighted volumetric strain rather than directly with hydrostatic stress defined using the bulk modulus.Source: §2.1 · pp. 7, 8, 10

2.2 Galerkin discretization and pressure elimination

The Galerkin mixed system can be reduced by eliminating pressure when its pressure block is invertible. Discontinuous pressure makes that elimination cell-local, but locality alone does not ensure stability.

2.2 Galerkin discretization and pressure elimination

  • For nearly incompressible solids, finite λ regularizes the pressure block and permits pressure elimination, unlike the fully incompressible limit.The displacement and pressure coefficients enter a Galerkin algebraic system that can be reduced by solving for pressure and substituting into the displacement row.Source: §2.2 · pp. 10–11
  • Discontinuous pressure makes the pressure block block-diagonal by pressure cell, so pressure elimination can be performed cell by cell.The resulting pressure Schur-complement operation is local, while the remaining system is displacement-only.Source: §2.2 · p. 11
  • A discontinuous pressure space does not by itself guarantee stability; the mixed space must satisfy the inf-sup condition or use consistent stabilization.Source: §2.2 · p. 11

2.3 Isogeometric approximation spaces

The section constructs B-spline and NURBS approximation spaces from knot vectors, then uses potentially different spline spaces for displacement and pressure in a mixed discretization.

2.3 Isogeometric approximation spaces

  • Interior-knot multiplicity determines spline continuity, while knot insertion permits spaces with different degrees, continuities, and mesh sizes.Simple interior knots give maximal continuity C^(k−1); repeating a knot k + 1 times produces a discontinuity.Source: §2.3 · p. 12
  • Tensor products extend the univariate spline construction to multiple dimensions, while the parametric-to-physical map uses control points and may use rational NURBS bases.The notation N_A denotes either a B-spline or NURBS basis function, depending on the geometric representation.Source: §2.3 · p. 12
  • Displacement and pressure fields use potentially different spline spaces, with separate polynomial degrees, continuities, and mesh sizes.These spaces are built within the same tensor-product B-spline framework.Source: §2.3 · p. 12
  • The mixed discretization pairs displacement and pressure spaces, with homogeneous displacement test functions vanishing on the essential boundary.Discontinuous pressure bases are broken across element interfaces, making the pressure block cellwise block diagonal and permitting local elimination.Source: §2.3 · p. 13

2.4 Inf–sup stability of the Galerkin formulation for isogeometric spaces

The discrete inf-sup condition governs pressure control in the incompressible limit, and spline continuity and relative mesh size affect stable pairings. The section contrasts continuous and discontinuous pressure choices and their stability and resolution trade-offs.

2.4 Inf–sup stability of the Galerkin formulation for isogeometric spaces

  • As incompressibility is approached, direct compressibility control weakens, so pressure stability must come from displacement–pressure coupling; poorly controlled modes can produce spurious pressure and stress oscillations.High continuity reduces locally independent displacement modes, making the relative richness of the displacement and pressure spaces important.Source: §2.4 · pp. 13–14
  • For uniformly inf-sup-stable pairs, the mixed error estimate remains controlled toward incompressibility, with balanced displacement and pressure approximation orders when ku = kp + 1 and mesh sizes are comparable.The error estimate relies on a uniform discrete inf-sup condition.Source: §2.4 · p. 15
  • The sufficient macroelement stability criterion assumes a single patch, aligned nested tensor-product meshes, quasi-uniform knots, and a regular geometry map; failure of the criterion alone does not prove instability.The condition is established under these hypotheses and applies to the structured spline pairs considered.Source: §2.4 · p. 16

2.4.1 Continuous pressure examples

  • The finite-element Taylor–Hood-type pair satisfies the stability count, but its maximally continuous IGA analogue fails the strict sufficient count despite the same one-degree difference.The discrete inf-sup tests establish lack of uniform stability for the maximally continuous analogue.Source: §2.4.1 · p. 17, Figure 2 · p. 18
  • Reducing displacement continuity by one order restores the sufficient count, while coarsening the pressure mesh by a factor of two provides another stable option that retains maximum displacement continuity.The coarser-pressure option is the subdivision pressure pair.Source: §2.4.1 · pp. 17–18

2.4.2 Discontinuous pressure spaces

  • For discontinuous pressure with full pressure accuracy and maximally continuous displacement, the sufficient criterion requires a pressure mesh coarsened by more than the displacement degree.The illustrated stable pair uses a pressure mesh four times as coarse, which may reduce pressure resolution and complicate assembly relative to a same-mesh local formulation.Source: §2.4.2 · p. 19, Figure 5 · p. 19
  • The motivation for stabilization is to retain discontinuous-pressure locality while avoiding severe coarsening, but stabilization does not remove the inf-sup requirement for elementwise constant pressure.Discontinuous pressure makes the pressure block cellwise block diagonal, allowing local pressure elimination.Source: §2.4.2 · p. 19

2.5 Residual-based stabilization

Residual-based stabilization adds elementwise momentum-residual terms to relax Galerkin inf-sup restrictions, but discontinuous pressures still require inf-sup control of their elementwise constant modes. Among the variants, VMS remains stable for every positive stabilization level, although very large levels can reduce accuracy.

2.5 Residual-based stabilization

  • The method adds consistent elementwise terms based on the strong momentum residual, which vanish when the exact solution satisfies the momentum equation.The parameter χ selects GLS, SUPG/PSPG-type, or VMS stabilization; GLS and SUPG/PSPG-type forms are conditionally stable with respect to the parameter.Source: §2.5 · pp. 20–21
  • VMS is unconditionally stable for any positive stabilization level, though very large levels can cause over-stabilization and loss of accuracy.Its stability follows from an identity that explicitly controls the pressure gradient.Source: §2.5 · pp. 21, 23
  • For discontinuous pressures, residual stabilization controls nonconstant elementwise modes but not constants, so the constant-pressure subspace must satisfy an additional inf-sup criterion.Elementwise constants have zero elementwise pressure gradient.Source: §2.5 · p. 23
  • With a fixed pressure-to-displacement mesh ratio, pressure-grid coarsening affects the error constant but does not change the asymptotic convergence rate.For ku = kp+1, displacement and pressure gradients attain their respective optimal approximation rates; more generally, the available estimate predicts a one-order loss for pressure-gradient convergence.Source: §2.5 · pp. 24–25
  • The available estimate predicts optimal displacement-gradient convergence but a one-order loss for pressure-gradient convergence; first-order equal-order approximations guarantee only boundedness of the pressure gradient.These conclusions are consistent with the classical residual-based Stokes analysis of Hughes et al..Source: §2.5 · p. 25

2.6 Stabilization parameter

The stabilization restrictions depend on the nondimensional product of the element parameter and the local inverse constant, rather than on the parameter alone. Normalizing by that constant makes the chosen stabilization level comparable across meshes, degrees, and element geometries.

2.6 Stabilization parameter

  • The relevant stabilization level is the nondimensional product te = τeΛe, where Λe is the local inverse constant.GLS requires 0 < te < 1, SUPG/PSPG-type requires 0 < te < 4 under the stated largest admissible Young parameter, and VMS is stable for any te > 0.Source: §2.6 · p. 25
  • The sharp local inverse constant is computed from a generalized eigenvalue problem.Λe depends on the local spline space, polynomial degree, element geometry, and quadrature domain.Source: §2.6 · p. 25
  • Choosing τe = t/Λe sets τeΛe = t on every element, normalizing the stabilization level across mesh sizes, polynomial degrees, and element geometries.Here t is the nondimensional stabilization level varied in the computations.Source: §2.6 · pp. 25–26

2.7 Pressure elimination in the stabilized formulation

The stabilized mixed system can be reduced to a displacement-only system by eliminating pressure locally, provided the stabilized pressure block is invertible. In nearly incompressible elasticity, finite compressibility supplies a genuine pressure block; the stabilization choice also determines whether the condensed operator is symmetric.

Pressure elimination

  • The stabilized mixed formulation reduces to a displacement-only system when its stabilized pressure block is invertible.For nearly incompressible elasticity, finite compressibility contributes a genuine pressure block, making exact pressure elimination possible under the stated stability conditions.Source: §2.7 · p. 26

Scope condition

  • In the fully incompressible Stokes case, stabilization may leave a pressure nullspace, so exact pressure elimination is not automatic.Here the Galerkin pressure block vanishes, while stabilization typically controls pressure-gradient modes.Source: §2.7 · p. 26

Local condensation

  • Because the pressure block is block diagonal by pressure cell, elimination can be performed cell by cell, leaving only displacement unknowns globally.The pressure field can then be recovered locally after solving for displacement.Source: §2.7 · pp. 26–27

Solver implications

  • The stabilization choice trades solver compatibility against parameter robustness: GLS gives a symmetric positive-definite condensed operator, whereas VMS is unconditionally stable but nonsymmetric.The stated solver options are conjugate gradients or Cholesky for GLS, and a nonsymmetric Krylov method for VMS.Source: §2.7 · p. 27

2.8 Stability condition for discontinuous pressure spaces

The stabilized formulation controls nonconstant pressure modes through pressure-gradient stabilization, but elementwise constant modes still require a discrete inf–sup condition. This permits a fixed factor-two pressure coarsening for maximally continuous displacement splines, independently of polynomial degree, or same-mesh stability by reducing displacement continuity by one order.

Pressure-control mechanisms

  • Pressure-gradient stabilization controls only the nonconstant pressure component; elementwise constants require separate control through displacement–pressure coupling.In the incompressible limit, the constant component is controlled through the inf–sup condition, while the nonconstant component is controlled by stabilization.Source: §2.8 · pp. 28–29
  • The two pressure components have complementary control: the constant component uses inf–sup stability, while the nonconstant component uses stabilization, keeping the incompressible-limit norm nondegenerate.For finite compressibility, an additional direct pressure-control term is supplied by the compressibility contribution.Source: §2.8 · p. 29

Stability analysis

  • The stabilization argument applies to spline spaces using an elementwise inverse estimate, constant-mode inf–sup stability, and coercivity; no pressure-jump stabilization is required.The cited proof ingredients do not depend on inter-element displacement continuity; the spline-specific condition is supplied by macroelement analysis.Source: §2.8 · p. 29

Same-mesh pair

  • The maximally continuous cubic same-mesh pair U(3, C2, h)/P(2, C−1, h) fails the sufficient constant-pressure condition, and discrete inf–sup tests confirm its instability.Higher-order discontinuous pressure spaces contain the same elementwise constant subspace, so pressure-gradient stabilization alone cannot provide the required uniform pressure control.Source: §2.8 · p. 30

Coarsen the pressure mesh

  • Coarsening the pressure mesh by a factor of two satisfies the constant-pressure macroelement condition while preserving maximum displacement continuity.The pressure space's full degree need not satisfy the remaining inf–sup condition; only its elementwise constant subspace must do so.Source: Coarsen the pressure mesh · p. 30, Figure 7 · p. 31

Reduce the displacement continuity

  • Alternatively, reducing displacement continuity by one order satisfies the constant-pressure condition on the same pressure mesh while retaining pressure-cell-local elimination.This provides the additional displacement richness needed for constant-mode stability.Source: Reduce the displacement continuity · p. 31, Figure 8 · p. 31

Comparison with the unstabilized formulation

  • Stabilization reduces the Galerkin coarsening requirement from n > k to n > 1, yielding fixed factor-two pressure coarsening independently of displacement polynomial degree.The unstabilized comparison uses kp = k −1 and r = k −1; the stabilized condition applies to the elementwise constant pressure subspace.Source: Comparison with the unstabilized… · pp. 31–32

Comparison of stable constructions

  • Pressure coarsening preserves maximum displacement continuity, avoids extra displacement degrees of freedom, reduces pressure unknowns, and retains pressure-cell-local condensation.Compared with the unstabilized Galerkin construction, the fixed factor-two mesh retains greater pressure resolution at higher polynomial degrees.Source: Comparison with the unstabilized… · p. 32

2.9 Immersed isogeometric discretization

The immersed isogeometric discretization separates the physical NURBS geometry from the background approximation grids and integrates over the physical domain using cut-cell quadrature, weak boundary enforcement, and stabilization. It uses extended B-splines for weakly supported boundary displacement functions and cellwise aggregation for discontinuous pressure degrees of freedom, expressing dependent boundary coefficients in terms of interior coefficients.

2.9 Immersed isogeometric discretization

  • Displacement and pressure bases lie on an independent background grid, while the weak form is integrated only over the physical domain and its boundary.The physical domain and boundary are described by a NURBS discretization, so the background analysis mesh need not conform to the physical boundary.Source: §2.9 · p. 32

2.9.1 Active mesh and cell classification

  • The active mesh comprises inner cells fully inside the domain and boundary cells intersected by its boundary; outer cells are inactive.Boundary-cell classification also identifies near-boundary regions where basis functions may have insufficient physical support, motivating displacement extension or pressure-cell aggregation.Source: §2.9.1 · p. 33

2.9.2 Subdivision quadrature for boundary cells

  • Boundary cells are integrated by recursive subdivision, with terminal cut regions approximated by polygons, triangulated, and integrated using Gaussian quadrature.Accepted inner subcells retain tensor-product Gaussian quadrature, while a one-dimensional surface rule integrates the physical boundary segment.Source: §2.9.2 · pp. 34–35

2.9.3 Nitsche method for essential boundary conditions

  • Essential boundary conditions are imposed weakly with Nitsche’s method because the physical boundary generally does not coincide with the background-grid boundary.The same formulation can be applied componentwise when only selected displacement components are prescribed; volume terms are evaluated over physical portions of active cells.Source: §2.9.3 · p. 36

2.9.4 Nitsche stability estimate

  • For cut cells, symmetric Nitsche requires γN,K > λN,K, whereas skew-symmetric Nitsche has a positive boundary contribution for any γN,K > 0.The skew-symmetric consistency contributions cancel in the mixed energy pairing; pressure stability remains governed by the inf-sup and residual-stabilization mechanisms.Source: §2.9.4 · p. 39

2.9.5 Cut-cell instability for continuous displacement fields

  • Small physical support can leave displacement control variables weakly constrained and severely worsen conditioning; an extended B-spline construction treats this cut-cell instability.The construction relies on B-spline polynomial reproduction: degree-k polynomials can be represented exactly in a degree-k basis.Source: §2.9.5 · p. 41
  • Boundary displacement coefficients are extrapolated from neighboring inner blocks with degree-k Lagrange coefficients, preserving polynomial completeness while removing weakly supported boundary basis functions.Continuous pressure uses the same extended B-spline construction; discontinuous pressure instead uses cellwise aggregation.Source: §2.9.5 · pp. 42, 44

2.9.6 Aggregation of discontinuous pressure degrees of freedom

  • Each cut boundary pressure cell is assigned to a nearest inner donor, whose polynomial is L2-projected over the full boundary cell to reconstruct dependent coefficients while preserving the local polynomial approximation.Boundary pressure coefficients are dependent on donor coefficients, so no cut cell retains independent pressure unknowns.Source: §2.9.6 · pp. 44–46
  • Aggregation is performed independently for each boundary cell, so it imposes no continuity across pressure-cell interfaces; local extension and aggregation maps are applied during assembly to reconstruct dependent coefficients.Source: §2.9.6 · pp. 47–48

3 Kirsch problem

The Kirsch benchmark assesses the accuracy, stability, and convergence of the mixed formulations near incompressibility, using analytical fields as references. It compares mesh-conforming and immersed discretizations across pressure-space classes.

3 Kirsch problem

  • The benchmark evaluates mixed-formulation accuracy, stability, and convergence using a circular-hole plate with a smooth analytical solution and strongly nonuniform stress and pressure fields.The setup probes locking in displacement-only formulations and possible spurious pressure and stress oscillations in inf-sup-unstable mixed formulations.Source: §3 · p. 48
  • The plate uses E = 100 MPa and ν = 0.49999 under plane-strain conditions, placing the material response very close to the incompressible limit.The hole radius is 1 mm and the applied far-field tensile stress is 1 MPa.Source: §3 · p. 48
  • Symmetry reduces the infinite plate to a quarter-annular domain truncated at rout = 4 mm, with a traction-free inner boundary and exact Kirsch traction on the outer boundary.That exact outer traction makes the restricted infinite-domain Kirsch solution exact for the finite quarter-annular problem.Source: §3 · pp. 48–49
  • The study compares mesh-conforming and immersed discretizations using continuous, discontinuous piecewise-constant, and higher-order discontinuous pressure spaces.The immersed geometry is embedded in an independent Cartesian background grid; its cut cells use recursive subdivision and terminal triangulation, with essential boundary conditions imposed weakly by Nitsche's method.Source: §3 · pp. 50–51

3.1 Continuous Pressure

The continuous- and constant-pressure studies compare mesh-conforming and immersed discretizations, testing stability, convergence, and the effects of VMS stabilization. They show that good benchmark convergence does not by itself establish inf-sup stability.

3.1.1 Mesh-Conforming

  • A pressure grid coarsened by a factor of two satisfies the inf-sup criterion, while both pressure-grid families show optimal convergence on the smooth mesh-conforming benchmark.The same-mesh pair is not certified, so its observed convergence is not evidence of uniform inf-sup stability.Source: §3.1.1 · p. 52, Figure 15 · p. 52
  • VMS stabilization reduces equal-order Galerkin errors and restores expected rates, but very large stabilization can over-diffuse pressure and increase errors.On a fixed 32 × 32 mesh, an intermediate stabilization range yields the smallest errors; small values leave instability effects, while excessive values cause over-stabilization.Source: §3.1.1 · pp. 53–54, Figure 16 · p. 53, Figure 17 · p. 54

3.1.2 Immersed discretization

  • In immersed continuous-pressure tests, both pressure-grid families retain optimal convergence, although the same-mesh pair remains uncertified and its errors are generally smaller for this benchmark.Cut configurations cause nonmonotone error variations, especially at higher orders, without changing the overall rates.Source: §3.1.2 · pp. 55–56, Figure 19 · p. 56
  • For immersed equal-order spaces, VMS stabilization reduces the larger Galerkin errors and restores regular convergence; stabilized pressure and stress fields are smooth despite cut cells.The comparison uses t = 5, and the field behavior is qualitatively consistent with the mesh-conforming case.Source: §3.1.2 · pp. 57–58, Figure 20 · p. 57

3.2.1 Mesh-Conforming

  • With piecewise-constant pressure, pressure and stress errors converge at approximately first order, and a 2h pressure grid maintains the constant-mode inf-sup condition across refinement.The same-mesh inf-sup constant decreases under refinement; discontinuities in the pressure approximation also appear as elementwise stress variations.Source: §3.2.1 · pp. 59–60, Figure 22 · p. 59

3.2.2 Immersed discretization

  • The immersed constant-pressure results show the same stable-versus-unstable inf-sup distinction, while retaining approximately first-order pressure and stress convergence.Cut-cell refinements add nonmonotone variations; for the smooth benchmark, mild pressure oscillations appear in the same-mesh field, and both fields show elementwise stress variation.Source: §3.2.2 · pp. 60–62, Figure 24 · p. 61

3.3 Higher-Order Discontinuous Pressure

The higher-order discontinuous-pressure studies test VMS stabilization with both same-mesh and coarsened pressure grids, then examine local parameter scaling in immersed cells. The results support the approach while identifying the constant-mode stability requirement and a geometric limitation of area-based normalization.

3.3 Higher-Order Discontinuous Pressure

  • VMS controls nonconstant elementwise pressure modes, but the underlying displacement-pressure pair must satisfy the inf-sup condition for the constant mode.For maximally continuous displacement splines, coarsening the pressure grid to 2h meets this condition; the same-mesh family does not.Source: §3.3 · p. 62

3.3.1 Mesh-Conforming

  • The same-mesh discontinuous-pressure family converges regularly on the smooth Kirsch problem despite lacking a uniform constant-mode inf-sup bound.Its constant unstable mode is not strongly excited in this benchmark, while VMS controls nonconstant modes; the low-error range is broad but excessive stabilization eventually increases errors, with the minimum shifting to larger t as polynomial order increases.Source: §3.3.1 · pp. 62–64
  • The coarsened 2h family satisfies the constant-mode inf-sup requirement and converges regularly, with an intermediate low-error stabilization range that becomes more localized as polynomial order increases.Both same-mesh and coarsened-pressure fields are smooth under VMS stabilization.Source: §3.3.1 · p. 64, Figure 28 · p. 65, Figure 29 · p. 65

3.3.2 Immersed discretization

  • In immersed discontinuous-pressure tests, both grid families retain overall convergence, but the same-mesh family’s slightly smaller errors do not change its constant-mode instability.Cut-cell configuration changes produce small nonmonotonic variations; the coarsened family retains the constant-mode inf-sup condition.Source: §3.3.2 · p. 66, Figure 31 · p. 67, Figure 32 · p. 67
  • Across the immersed higher-order discontinuous-pressure tests, VMS stabilization and pressure-cell aggregation retain effective, robust behavior in the presence of cut cells.The computed fields remain smooth at the macroscopic scale and qualitatively consistent with mesh-conforming results.Source: §3.3.2 · p. 66

3.4 Local stabilization-parameter scaling

  • Area-based normalization does not account for slender cut-cell aspect ratios, so the local stabilization parameter can become very small.For a strip-like cut region of width δ, τK scales as O(δ^2/(2µ)).Source: §3.4 · p. 68

4 Cook’s Membrane

Cook’s membrane tests mixed formulations under nearly incompressible, strongly nonuniform pressure and stress fields, in both mesh-conforming and immersed settings. Across the pressure families, the results distinguish unstable same-mesh pairs from stable 2h pairs and show how VMS controls nonconstant pressure modes while constant modes require inf-sup control.

4.1.1 Mesh-Conforming

  • For continuous-pressure spaces, the mesh-conforming same-mesh Galerkin pair oscillates, whereas pressure coarsening to 2h and VMS stabilization produce smooth fields.The 2h Galerkin pair is inf-sup stable; VMS controls unstable continuous pressure modes in the same-mesh pair.Source: §4.1.1 · p. 71, Figure 34 · p. 72
  • For continuous-pressure stabilization, GLS and SUPG/PSPG-type errors rise beyond a useful stabilization range, whereas VMS retains a broad stable range but can lose accuracy when over-stabilized.VMS stability with positive t does not make its accuracy parameter-independent.Source: §4.1.1 · pp. 71–72, Figure 42 · p. 79

4.1.2 Immersed

  • In the immersed continuous-pressure case, the same-mesh Galerkin pair oscillates, but the coarsened Galerkin and VMS pairs produce smooth fields without added cut-cell oscillations.The immersed inf-sup results show a mesh-independent constant for the 2h family, while same-mesh and equal-order families deteriorate under refinement.Source: §4.1.2 · pp. 72–73, Figure 35 · p. 73, Figure 36 · p. 74

4.2 Constant Pressure

  • For constant-pressure spaces, coarsening the pressure grid from h to 2h satisfies the strict constant-mode inf-sup criterion for every tested displacement order.The same-mesh criterion fails; immersed refinement results likewise show decreasing βh for h and an approximately mesh-independent constant for 2h.Source: §4.2 · p. 74, §4.2.2 · p. 76, Figure 40 · p. 77

4.2.1 Mesh-Conforming

  • In the mesh-conforming constant-pressure fields, the same-mesh pair shows pressure oscillations, while the 2h pair gives a more regular pressure distribution.Stress retains element-scale variation for both pairs because the pressure approximation is discontinuous and piecewise constant.Source: §4.2.1 · pp. 74–75, Figure 38 · p. 75

4.2.2 Immersed

  • In immersed constant-pressure fields, the same-mesh pair oscillates while the 2h pair is more regular; pressure-cell aggregation and cut-cell treatment preserve constant-mode stability.Stress variations are expected for both because the pressure approximation is discontinuous.Source: §4.2.2 · p. 76, Figure 39 · p. 76

4.3.1 Mesh-Conforming

  • For mesh-conforming higher-order discontinuous pressure, VMS improves both pressure-grid variants, but the 2h pair is smoother because it also controls the constant mode.VMS acts on nonconstant pressure modes; the constant mode remains more uncontrolled in the same-mesh pair.Source: §4.3.1 · pp. 77–78, Figure 41 · p. 78

4.3.2 Immersed

  • In the immersed higher-order discontinuous case, the 2h VMS pair produces smooth pressure and stress fields, including near cut cells, while the same-mesh pair retains pressure oscillations.The same-mesh oscillations are associated with insufficient inf-sup control of the elementwise constant pressure mode.Source: Figure 43 · p. 79, §4.3.2 · p. 80

5 Conclusions

The conclusions identify separate controls for discontinuous pressure modes and report that the immersed extension preserves the stability properties of the underlying pairs.

5 Conclusions

  • VMS controls nonconstant elementwise pressure modes, while the constant mode requires inf-sup compatibility; a pressure grid coarsened by two is sufficient for all polynomial orders considered.This relaxes the degree-dependent coarsening required by the corresponding Galerkin formulation while retaining maximum displacement continuity and local pressure elimination.Source: §5 · p. 80
  • Element-local discontinuous pressure support permits static condensation, leaving a globally coupled displacement-only system and allowing local pressure recovery.Source: §5 · p. 80
  • The numerical studies of the Kirsch and Cook’s membrane problems support the stability analysis; Cook’s unstable pairs show pronounced pressure and stress oscillations, while stable 2h families remain well behaved.Source: §5 · p. 80
  • GLS preserves symmetry of the condensed displacement operator for conjugate-gradient or Cholesky solvers but has a conditional stabilization bound; VMS is unconditionally stable but nonsymmetric.The authors favor GLS when solver reuse dominates and VMS when robustness to the stabilization parameter dominates.Source: §5 · p. 81
  • The immersed extension uses extended B-splines, pressure-cell aggregation, subdivision-based cut-cell quadrature, and Nitsche enforcement; its inf-sup studies and solutions indicate preserved stability with nonconforming cut cells.Source: §5 · p. 81

Appendix A Numerical evaluation of the discrete inf–sup constant

The appendix evaluates discrete inf-sup constants numerically through a generalized eigenvalue procedure. Refinement behavior indicates whether a displacement–pressure family remains uniformly stable.

Appendix A Numerical evaluation of the discrete inf–sup constant

  • A generalized eigenvalue procedure evaluates the discrete inf-sup constant over uniformly refined meshes to test whether βh remains bounded away from zero as h approaches zero.Source: Appendix A · pp. 81, 83
  • The displacement stiffness matrix defines the discrete displacement-energy norm, equivalent to the H1 norm when homogeneous essential boundary conditions eliminate rigid-body modes.Source: Appendix A · p. 81
  • The pressure Schur complement leads to a generalized eigenvalue problem whose smallest positive eigenvalue gives the square of the discrete inf-sup constant.The displacement space is restricted to eliminate rigid-body modes.Source: Appendix A · pp. 81–82
  • The test excludes finite-compressibility and stabilization contributions, so βh measures intrinsic LBB compatibility of the selected spaces.Source: Appendix A · p. 82
  • For immersed tests, the displacement norm includes the Nitsche boundary contribution, and the matrices incorporate the extension, aggregation, cut-cell quadrature, and weak boundary enforcement.The norm is positive definite when the Dirichlet boundary has sufficient measure to eliminate rigid motions.Source: Appendix A · p. 82
  • A fitted slope near zero indicates an approximately mesh-independent inf-sup constant, whereas a positive slope indicates deterioration under refinement.Source: Appendix A · p. 83
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