GMCP: Geometric Mortar Contact Potential

• GMCP is a contact formulation combining mortar integration and barrier functions to achieve spatially accurate contact pressures, improving over pointwise IPC discretizations.
• It employs geometrically constructed samples via projection and clipping with adaptive support radii and stiffness parameters for robust non-penetration enforcement.
• The framework demonstrates patch-test accuracy and reliable performance in tactile sensing and isogeometric analyses, addressing nonmatching interface challenges.

Geometric Mortar Contact Potential (GMCP) denotes a contact formulation that combines mortar-style interface integration with a barrier-based non-penetration model. In the 2026 IsaacIPC framework, GMCP is introduced for tactile sensing as a barrier potential over contact samples on tactile surfaces, with the explicit goal of resolving contact-pressure distributions more accurately than pointwise IPC sampling on discrete geometric primitives (Liang et al., 23 May 2026). In the broader mortar literature, the 2020 review "Frontiers in Mortar Methods for Isogeometric Analysis" does not use the acronym GMCP, but it presents a geometric, variationally consistent mortar contact formulation on exact NURBS/B-spline geometry that embodies the same core idea: contact potentials and constraints defined on the interface geometry and enforced through mortar operators, with robust treatment of large sliding, large deformation, and nonmatching discretizations (Hesch et al., 2020).

1. Terminology, scope, and lineage

GMCP emerged in a setting where tactile sensing requires not only robust non-penetration but also an accurate and spatially consistent contact-pressure distribution on the tactile surface. IsaacIPC positions GMCP against conventional IPC discretizations whose barrier potentials are attached to pointwise geometric primitives such as vertex–face and edge–edge interactions; that discretization is robust for large-deformation dynamics but can exhibit pressure aliasing and poor pressure transfer across nonmatching interfaces, which is particularly problematic when the signal of interest is an integrated traction field on a tactile pad (Liang et al., 23 May 2026).

The mortar background comes from finite element contact and multi-patch coupling. Mortar methods enforce interface conditions in a weak, integrated form across overlapping slave–master regions, and in the isogeometric setting they operate directly on spline parameterizations of the interface. The 2020 review emphasizes that this is especially relevant for complex geometries composed of multiple patches, where coupling technologies must preserve the advantages of isogeometric analysis, including exact CAD geometry and higher-order discretizations (Hesch et al., 2020).

The phrase geometric mortar has a closely related but context-dependent meaning in the two sources. In the IsaacIPC formulation, it refers to constructing interface quadrature geometrically by projecting master triangles, edges, and vertices onto the tangent plane of a slave triangle and clipping them to obtain overlap domains. In the isogeometric mortar literature, geometric exactness refers to evaluating contact quantities on exact NURBS/B-spline surfaces, so that gaps and normals are computed directly on the CAD geometry rather than on an approximated surface. This suggests that GMCP is best understood as a family of contact constructions in which the interface geometry, rather than a nodewise surrogate, is the primary object of discretization.

2. Variational structure

In the isogeometric mortar setting, the starting point is the total potential energy for constrained elasticity. The bulk part is given by

$$\Pi_{\mathrm{bulk}}(u) = \int_{\mathcal{B}_0} \Psi(F(u))\,\mathrm{d}V - \int_{\mathcal{B}_0} B \cdot u\,\mathrm{d}V - \int_{\Gamma^n} T \cdot u\,\mathrm{d}A,$$

with $F(u)=\nabla u$. The contact contribution is written abstractly as

$$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$

where $g_n$ and $g_t$ are the normal and tangential gaps and $\lambda$ is the mortar Lagrange multiplier field. For frictionless normal contact, an archetypal potential is

$$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$

which encodes the Kuhn–Tucker conditions

$$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$

Penalty, Nitsche, and augmented-Lagrangian variants are also part of the same variational landscape (Hesch et al., 2020).

IsaacIPC reformulates the contact part as a discrete barrier energy rather than a multiplier formulation. Let $K_q$ denote the samples of type $q \in \{2,1,0\}$, corresponding to face, edge, and point samples. GMCP defines

$F(u)=\nabla u$0

where $F(u)=\nabla u$1 is the stiffness for a sample type, $F(u)=\nabla u$2 is the geometric quadrature weight, $F(u)=\nabla u$3 is a master-feature transition weight, $F(u)=\nabla u$4 is an adaptive barrier support radius, and $F(u)=\nabla u$5 is an IPC-style barrier function that diverges as $F(u)=\nabla u$6 and vanishes for $F(u)=\nabla u$7. The formulation is explicitly described as following IPC in its barrier mechanics while replacing IPC’s pointwise contact discretization with a mortar-style, geometrically integrated barrier over slave-side samples (Liang et al., 23 May 2026).

The first variation gives a direct contact-pressure interpretation. At a sample $F(u)=\nabla u$8,

$F(u)=\nabla u$9

so the normal traction magnitude is identified as

$$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$0

For face samples, $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$1 is a pressure; for edge and point samples, it acts as a line or point contribution that is distributed to slave nodes through barycentric weights (Liang et al., 23 May 2026).

A central distinction between the two strands is therefore enforcement. Classical mortar contact in IGA often uses dual Lagrange multipliers, penalty, Nitsche, or augmented-Lagrangian enforcement. GMCP in IsaacIPC imports the mortar integration pattern but keeps the feasible-iterate, barrier-based contact model of IPC (Liang et al., 23 May 2026).

3. Interface geometry, gaps, and sample construction

In the isogeometric mortar setting, a master surface $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$2 and a slave surface $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$3 are given by spline parameterizations $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$4 and $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$5. The normal gap is

$$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$6

with $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$7, and the tangential gap is

$$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$8

These quantities are evaluated on exact NURBS surfaces, which is the basis for the geometric exactness emphasized in isogeometric contact (Hesch et al., 2020).

In GMCP for tactile surfaces, the slave surface $$\Pi_{\mathrm{contact}}(u,\lambda) = \int_{\Gamma_c} \Phi\big(g_n(u),g_t(u),\lambda\big)\,\mathrm{d}\Gamma,$$9 is triangulated, and each slave triangle $g_n$0 carries a unit normal $g_n$1, taken per triangle and used as the contact normal. The master surface $g_n$2 contributes features of dimension two, one, and zero: triangles $g_n$3, edges $g_n$4, and vertices $g_n$5. Contact samples are constructed by projecting these master features onto the tangent plane of a slave triangle and clipping the result against the slave triangle (Liang et al., 23 May 2026).

The three sample classes are:

• Face samples ($g_n$6): generated from the 2D overlap polygon between a master triangle and a slave triangle projected to the slave tangent plane and clipped by Sutherland–Hodgman; the polygon is fan-triangulated and integrated with standard 2D Gauss quadrature.
• Edge samples ($g_n$7): generated by projecting a master edge onto the slave tangent plane and clipping the resulting segment by the triangle; integration uses 1D Gauss quadrature along the clipped segment.
• Point samples ($g_n$8): generated by projecting a master vertex onto a slave triangle.

For any sample $g_n$9, GMCP stores a slave point $g_t$0, an associated master point $g_t$1, the unit slave normal $g_t$2, a scalar weight $g_t$3, and the signed normal gap

$g_t$4

The points are assembled from barycentric interpolants. For face samples,

$g_t$5

For edge samples,

$g_t$6

For point samples, $g_t$7 is the master vertex and $g_t$8 is its projection on $g_t$9 (Liang et al., 23 May 2026).

GMCP further smooths feature transitions through a $\lambda$0 Hermite step $\lambda$1. For face samples, $\lambda$2; for edge samples, $\lambda$3. The master-feature transition weight is

$\lambda$4

This tapers face contributions near master-triangle edges and edge contributions near edge endpoints, while vertex contributions remain full-weight (Liang et al., 23 May 2026).

To prevent nonphysical far-field forces, the adaptive support radius is chosen as

$\lambda$5

where $\lambda$6 is the current gap to the associated master feature. The stated purpose is to keep $\lambda$7, suppress rest-state forces, and retain a bounded activation region near contact (Liang et al., 23 May 2026).

4. Discretization, quadrature, and solution algorithms

Classical mortar methods introduce weak interface constraints through an $\lambda$8-type projection between slave and master traces. In the isogeometric review, the interface functional for displacement continuity is written as

$\lambda$9

and in contact this becomes the weak non-penetration condition

$$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$0

The discretized counterpart uses spline bases for both $$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$1 and $$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$2. Dual mortar spaces are designed to satisfy biorthogonality,

$$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$3

with partition of unity and local support enabling local static condensation of multipliers. The same review emphasizes that uniformly stable pairings require careful degree matching; for example, primal $$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$4 with dual $$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$5 can be unstable and produce spurious oscillations, whereas appropriately constructed dual spaces with crosspoint modifications satisfy the uniform inf–sup condition (Hesch et al., 2020).

GMCP does not use dual multipliers, but it inherits the mortar emphasis on interface quadrature over overlap domains. Its gradient and Gauss–Newton-style Hessian approximation are

$$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$6

$$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$7

with the Hessian projected to the positive semidefinite cone. Using the per-triangle constant normal $$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$8, the positional derivatives are localized and sparse; for a face sample,

$$\Phi(g_n,\lambda_n)=\lambda_n g_n + I_{(-\infty,0]}(\lambda_n),$$9

with analogous expressions for edge and point samples. The normal variation within $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$0 is neglected, which is stated to be consistent with per-triangle tangent-plane projection and standard segment-to-segment or element-to-surface mortar practice (Liang et al., 23 May 2026).

The algorithmic pipeline in IsaacIPC is organized per time step as broad phase, projection and clipping, quadrature and sampling, gap and normal evaluation, master-feature transition weighting, adaptive support selection, barrier evaluation, gradient/Hessian assembly, GPU solve, and line search with a gap safeguard. If a candidate displacement $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$1 decreases a gap, GMCP uses

$$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$2

and then takes the global step size $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$3, so that all gaps remain positive during line search (Liang et al., 23 May 2026).

The isogeometric literature adds complementary algorithmic ingredients not currently present in normal-only GMCP: consistent linearization of closest-point projections, semi-smooth Newton methods, primal-dual active set strategies, thermomechanical interface constraints, and energy-momentum consistent time integration for dynamic contact. This suggests that the algorithmic space surrounding GMCP is broader than barrier mechanics alone, even though IsaacIPC adopts the IPC-compatible branch of that space (Hesch et al., 2020).

5. Mechanical properties and numerical behavior

The isogeometric mortar literature attributes several desirable properties to variationally consistent geometric mortar discretizations: energy consistency, frame indifference, momentum conservation in conjunction with energy-momentum consistent time integrators, patch-test correctness, robustness for large sliding and finite strains, and modular coupling across nonmatching meshes. For contact with limited regularity, the reported asymptotic rate in the energy norm is typically $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$4, even with higher-order NURBS; dual mortar bases that do not reproduce the full higher order can still attain that rate, which is why they remain attractive for contact (Hesch et al., 2020).

In IsaacIPC, GMCP is motivated by an analogous goal but evaluated through contact-pressure fidelity on tactile surfaces. Because the barrier is integrated over slave-side overlap samples rather than tied to isolated primitive contacts, the reported effect is improved consistency across nonmatching meshes and suppression of pressure aliasing. The framework also states that denser face and edge sampling improves pressure smoothness and patch-test performance (Liang et al., 23 May 2026).

The reported contact patch test uses two elastic blocks of size $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$5 with a $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$6 initial gap and mixed resolutions of $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$7 versus $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$8 tetrahedra, in a quasi-static frictionless setting with $$g_n \ge 0,\qquad \lambda_n \le 0,\qquad \lambda_n g_n = 0.$$9 and target $K_q$0. The numerical results are:

$K_q$1	Max relative error in $K_q$2	Max spurious stress $K_q$3
$K_q$4	$K_q$5	$K_q$6
$K_q$7	$K_q$8	$K_q$9
$q \in \{2,1,0\}$0	same as $q \in \{2,1,0\}$1	same as $q \in \{2,1,0\}$2

These results are reported as evidence that mortar-style integration produces accurate, stable pressure transfer across nonmatching interfaces and passes the patch test to high accuracy (Liang et al., 23 May 2026).

For smooth curved contact, IsaacIPC reports a Hertzian benchmark consisting of one-eighth of a hemisphere of radius $q \in \{2,1,0\}$3 indenting a block sector, with both bodies linearly elastic $q \in \{2,1,0\}$4, quasi-static and frictionless, $q \in \{2,1,0\}$5, and the load ramped in $q \in \{2,1,0\}$6 steps. The computed normal pressure distribution is said to match the Hertz solution

$q \in \{2,1,0\}$7

with small oscillations that decrease under mesh or quadrature refinement (Liang et al., 23 May 2026).

The isogeometric review reports a related 2D Hertzian contact example summarized from Seitz et al., where under uniform refinement for $q \in \{2,1,0\}$8 and $q \in \{2,1,0\}$9 NURBS, both standard and dual Lagrange multiplier bases achieve $F(u)=\nabla u$00 convergence in the energy norm. For thermomechanical contact, a two-body frictionless contact problem with thermal coupling and Neo-Hookean materials is reported to show $F(u)=\nabla u$01 convergence of displacement and temperature $F(u)=\nabla u$02 semi-norms, with IGA competitive or slightly better for displacement errors when plotted against the number of control points or nodes (Hesch et al., 2020).

6. Applications, limitations, and open directions

Within IsaacIPC, GMCP is integrated into a robotic simulation stack that couples GPU-accelerated IPC in libuipc with IsaacSim/Lab. All contact sampling, barrier evaluation, and solver steps run on the GPU, and simulated deformations are transferred from the simulation mesh to visual and tactile meshes through stored barycentric and normal-offset embeddings. Reported qualitative demonstrations include a quadruped with soft foot pads, a dexterous hand with green elastomer fingertips on a $F(u)=\nabla u$03-DoF hand, and a UMI gripper whose inner pads contact a rigid cube and a softer cylinder. In these examples, GMCP is described as producing smooth or localized pressure maps appropriate for tactile rendering, policy evaluation, and parallel rollout logging (Liang et al., 23 May 2026).

The present IsaacIPC formulation is explicitly normal-only. Tangential traction, friction cones, stick–slip, and shear coupling are listed as future work, and no frictional gradient or Hessian terms are included in $F(u)=\nabla u$04. This is a major point of departure from the more general mortar literature, where frictional contact is already formulated through Coulomb constraints,

$F(u)=\nabla u$05

with stick and slip conditions, and where thermomechanical contact includes a thermal multiplier $F(u)=\nabla u$06, contact heat conductivity $F(u)=\nabla u$07, heat partition parameter $F(u)=\nabla u$08, and objective tangential relative velocity $F(u)=\nabla u$09 through

$F(u)=\nabla u$10

(Hesch et al., 2020).

Both lines of work identify geometric and numerical challenges. In IsaacIPC, geometric projection and clipping become intricate near sharp features or rapidly changing topologies; sampling density and the transition widths $F(u)=\nabla u$11 affect smoothness versus locality; and very stiff $F(u)=\nabla u$12 can lead to ill-conditioning, motivating preconditioning and barrier–AL hybridization, with Barrier-AL and AL-IPC named as promising directions (Liang et al., 23 May 2026). In isogeometric mortar methods, open issues include inf–sup constraints and dual space selection, locking or overconstraint without crosspoint or wirebasket modifications, quadrature sensitivity on curved interfaces and triple integrals, and nontrivial HPC assembly of transfer operators on nonmatching meshes. The review also identifies weak $F(u)=\nabla u$13 coupling, $F(u)=\nabla u$14 shell coupling, multidimensional coupling, thermomechanical and fracture coupling, and a fully potential-based frictional GMCP with temperature dependence and large sliding on complex geometries as active topics (Hesch et al., 2020).

Taken together, these sources place GMCP at the intersection of two traditions: mortar-based weak interface discretization and IPC-style barrier mechanics. In the tactile simulation context, GMCP is a discrete barrier energy over geometrically constructed slave-side samples. In the isogeometric contact context, the same label naturally refers to a geometric, variationally consistent mortar contact potential posed directly on exact spline geometry. The common denominator is that contact is resolved through integrated interface geometry rather than purely pointwise primitives, with the contact pressure field treated as a first-class numerical quantity rather than a by-product (Liang et al., 23 May 2026).