Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)

Elena Atroshchenko; Gang Xu; Satyendra Tomar; Stephane P. A. Bordas

Source-linked AI summary

Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)

Elena Atroshchenko, Gang Xu, Satyendra Tomar, Stephane P. A. Bordas

arXiv:1706.06371v2Jun 20, 2017math.NAcs.NA

TL;DR

IGA’s shared geometry and solution spaces can make local field refinement awkward when CAD geometry is represented by NURBS. GIFT uses separate geometry and field spaces, and the paper reports optimal-order convergence in tested cases even when the standard patch test fails.

Problem
NURBS-based IGA ties geometry and solution approximation together, although local field refinement may be needed while retaining a coarser geometry.
Source: §1.3 · p. 2, §1.4 · p. 3
Method
GIFT retains the geometry representation while approximating the solution in a separate field space that can be selected independently.
Source: §1.4 · p. 3, §1.6 · p. 5
Results
All reported cases exhibit optimal-order convergence, including cases that fail the classical patch test.
Source: §5 · p. 27
Takeaways & Limitations
GIFT allows local refinement of the solution field without modifying the geometry parameterization, and the paper gives recommendations for discretization choices yielding optimal convergence.
Source: §6 · p. 30
The reported convergence exception occurs when a singular point lies inside a solution element; the authors recommend avoiding combinations that violate geometry-parametrization continuity in solution elements.
Source: §5 · p. 27

Abstract

from arXiv · show

This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution field. The method inherits the main advantage of isogeometric analysis, i.e. preserves the original, exact CAD geometry (for example, given by NURBS), but allows pairing it with an approximation space which is more suitable/flexible for analysis, for example, T-splines, LR-splines, (truncated) hierarchical B-splines, and PHT-splines. This generalization offers the advantage of adaptive local refinement without the need to re-parameterize the domain, and therefore without weakening the link with the CAD model. We demonstrate the use of the method with different choices of the geometry and field splines, and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for non-nested spaces.

1 Introduction

IGA links CAD geometry and solution approximation through the same shape functions, but NURBS-based local refinement can require geometry re-parameterization. GIFT separates the geometry and field spaces, retaining the CAD geometry while allowing flexible field approximation and refinement.

1.1 Contribution; 1.6 Generalizing the isogeometric concept

GIFT separates geometry and solution spaces, preserving exact CAD geometry while allowing locally refined field approximations without refining the geometry.The paper presents NURBS geometry paired, for example, with PHT-spline field approximation, and identifies simpler polynomial approximations as a potential computational saving.Source: §1.1 · p. 1, §1.6 · p. 5

1.2 Background

The introduction situates IGA as a CAD-linked spline method and notes related approaches including implicit or immersed boundaries, subdivision surfaces, and NURBS-enhanced finite elements.The paper identifies subdivision surfaces as a unified modeling-and-simulation representation and NURBS-enhanced elements as using CAD geometry near boundaries.Source: §1.2 · p. 2

1.3 NURBS: Advantages and limitations; 1.4 Motivation

NURBS support exact conic representation and high continuity, but their tensor-product structure limits local refinement and can complicate watertight multi-patch geometry.The paper notes that local field refinement may be needed while retaining a coarser geometry, including for singularities, boundary layers, and steep gradients.Source: §1.3 · p. 2, §1.4 · p. 3

1.4 Motivation

IGA uses the same splines for geometry and solution, so changing CAD geometry carries into the field approximation without regenerating the mesh at each design iteration.The paper motivates decoupling because locally refining NURBS geometry may require re-parameterization, consume time, and risk loss of geometric exactness.Source: §1.4 · pp. 2–3

1.5 Alternatives to NURBS; 1.5.1 Hierarchical B-splines and Truncated Hierarchical B-splines (THB-splines); 1.5.2 T-splines; 1.5.3 PHT-splines

Alternatives to NURBS offer local-refinement options: THB-splines modify hierarchical B-splines, T-splines preserve exact NURBS geometry, and PHT-splines offer simple refinement with C1 continuity.The text also notes that general T-splines may lack linear independence, while polynomial PHT-splines may not represent arbitrary CAD geometry exactly.Source: §1.5.1 · p. 4, §1.5.2 · p. 4, §1.5.3 · p. 4

1.5.4 Geometric design and IGA considerations for unstructured quadrilateral meshes

Prior work on unstructured quadrilateral meshes constructs smooth, locally supported splines, but refinement can cause slight geometry variations.That approach defines design and analysis spaces through the spline spans and provides basis-change rules for isogeometric compatibility.Source: §1.5.4 · p. 5

1.6 Generalizing the isogeometric concept

GIFT permits distinct design and analysis spaces, including cases where the design space is not contained in the analysis space.The paper’s implementation uses NURBS geometry with NURBS, PHT-spline, or B-spline solution approximations and examines whether selected combinations pass the patch test.Source: §1.6 · p. 5
The paper presents GIFT as a proof of concept and reports that optimal convergence can occur for non-nested geometry and field spaces despite failure of the standard patch test.The introduction frames the numerical examples around different geometry and solution bases and their patch-test behavior.Source: §1.1 · p. 1, §1.6 · p. 5

2 Formulation of GIFT

GIFT maps a parametric domain to the physical domain using a geometry basis, while approximating the solution with a potentially different field basis. The formulation evaluates field derivatives through the geometry map and inserts the resulting approximation into the weak form to obtain a linear system.

2 Formulation of GIFT

The domain is mapped from a parametric domain to the physical domain, with the geometry basis defining the map and its Jacobian.The formulation introduces multi-index notation for the geometry basis and its basis functions.Source: §2 · p. 6
GIFT departs from classical IGA by choosing a solution basis that differs from the geometry basis and representing the solution with its unknown control variables.Field derivatives with respect to physical coordinates are evaluated using the chain rule and the inverse-transpose Jacobian.Source: §2 · p. 7
Substituting the solution approximation into the weak form yields a linear system with stiffness matrix, force vector, and unknown control variables.The paper gives the system assembly in matrix form after defining the weak form.Source: §2 · p. 7
The numerical examples use Poisson’s equation and linear elasticity to demonstrate the formulation.The paper briefly describes the weak-form derivations for both problems.Source: §2 · p. 7

2.1 Poisson’s equation

For Poisson’s problem, the weak form includes a source function and prescribed flux on the Neumann boundary.The resulting discrete equations are specified for indices i and j from 1 to M.Source: §2.1 · p. 7

2.2 Linear elasticity

For two-dimensional linear elasticity, the weak form uses displacement, body-force, and traction variables with plane-strain material parameters.The strain operator acts on the solution approximation to produce the stiffness matrix and force vector; solution derivatives use the geometry-based chain rule.Source: §2.2 · pp. 7–8

3 Patch tests

The patch tests compare geometry and solution bases on a quarter annulus, using Laplace and linear-elasticity problems. Across the tested pairings, passing requires geometry and field bases equivalent up to degree elevation or knot insertion, while some non-equivalent pairings fail.

3.1 Problem definition

The problems are Laplace and linear elasticity on a quarter-annulus domain, with linear solutions specified for both.Source: §3.1 · pp. 8–9

3.2 Geometry parametrization

The geometry cases include coarsest NURBS parametrization Q0, uniform knot-inserted refinements, and a non-uniform parametrization with arbitrarily chosen inner control points and weights.Source: §3.2 · pp. 9–10

3.3 Results and discussion of patch tests

In T 0, the first two basis combinations pass and the other two fail; in T 1, all three combinations pass.Source: §3.3 · p. 12
All three combinations pass in T 2 with different uniform geometry parameterizations, and in T 3 with non-uniform parameterization C1 and its degree-elevated version C2.Source: §3.3 · pp. 14–15
T 4 fails for all combinations of A1/A2 solution bases with C1/C2 geometry because the analogous relation between those bases does not hold.Source: §3.3 · p. 15
All T 5 combinations fail; this practical pairing uses NURBS A1 for geometry and B-splines D_i for solution, which cannot exactly represent the geometry.Source: §3.3 · p. 15
Geometry and field bases equivalent up to degree elevation or knot insertion are sufficient for the patch test to pass.Source: §3.3 · p. 15

4 Numerical examples

The numerical examples cover Laplace problems and linear elasticity in two and three dimensions, with varied geometry and solution bases. Across these tests, convergence generally follows the solution-basis degree, and PHT-spline results illustrate the benefits and trade-offs of weakly coupling geometry and simulation.

4.1.1 Example 1

In the quarter-annulus Laplace test, convergence follows the lowest solution-basis degree across equivalent geometry parameterizations, with only slight differences for non-uniform geometry.Even when the B-spline solution basis cannot exactly represent the geometry, the tested cases show expected convergence rates despite failing the patch test.Source: §4.1.1 · p. 16, Figure 8 · p. 16, Figure 9 · p. 17

4.1.2 Example 2

In example 4.1.2, with geometry fixed as N1,4, convergence rate depends only on the lowest solution-approximation degree, and NURBS and B-spline solution-basis results are almost identical.The geometry is fixed as a NURBS basis of degrees pξ = 1 and pη = 4; tested solution bases vary in degree and type.Source: §4.1.2 · p. 17, Figure 12 · p. 19

4.2.1 Example 1 (pressurized cylinder)

For the pressurized-cylinder elasticity test, convergence rate is governed by the lowest solution-basis degree, independent of the exact geometry parameterization.The reported results follow the pattern observed in the quarter-annulus Laplace example.Source: §4.2.1 · p. 20, Figure 14 · p. 20, Figure 15 · p. 21

4.2.2 Example 2 (plate with a circular hole)

For the plate-with-hole test, convergence depends on solution-basis order; bases of the same order produce almost identical results across the tested choices.The geometry parameterization remains fixed during refinement, with additional tests changing its interior weights.Source: §4.2.2 · p. 22, Figure 18 · p. 24

4.3 Linear elasticity in three-dimensions

In the pressurized-sphere test, convergence rate across all five cases depends only on the numerical solution approximation basis.The example models one-eighth of a hollow sphere using symmetry boundary conditions.Source: §4.3 · pp. 22, 25, Figure 20 · p. 26

4.4 Numerical solution with PHT splines

Locally refined PHT-spline cases converge faster than tensor-product bases; with cubic bases, GIFT B-splines and IGA NURBS have the same convergence rate.For the simple domain tested, exact geometry adds only a minor advantage because cubic PHT-splines approximate it below the discretization error; using coarse exact NURBS avoids repeated CAD communication and geometry refinement and storage.Source: §4.4 · pp. 26–27

5 Discussion on numerical results

The discussion distinguishes patch-test passage from optimal-order convergence: the tested combinations can converge optimally despite failing the classical patch test. It also cautions that basis choices must preserve continuity of the geometry parameterization within solution elements.

5 Discussion on numerical results

The tested geometry-solution combinations achieve optimal-order convergence even when they fail the classical patch test, provided the solution basis is suitable and geometry continuity is preserved within solution elements.The discussion notes a counterexample with a singular point inside a solution element that does not show the expected convergence rate, and recommends avoiding combinations that violate this continuity condition.Source: §5 · p. 27

5.1 Naming convention

The proposed naming convention classifies cases by geometry-basis degree p_g, while p_u denotes the solution-basis degree.The paper contrasts this geometry-centered naming with standard finite-element naming, which is based on p_u.Source: §5.1 · p. 27
With exact geometry representation, super-parametric approximations (pu < pg) can also deliver optimal orders of convergence; standard FEM does not recommend this case [71, P.172].Source: §5.1 · p. 27

6 Conclusions

GIFT relaxes the coupling between geometry and field spaces while retaining geometric exactness and direct CAD operation. Recommendations for choosing discretizations yield optimal convergence, and local refinement can capture sharp gradients without changing the geometry parameterization.

6 Conclusions

GIFT relaxes the coupling between geometry representation and field approximation while retaining geometric exactness and direct operation from CAD files.Source: §6 · pp. 27, 30
The recommended relative choices of geometry and field discretizations yield optimal convergence.Source: §6 · p. 30
Local refinement can capture sharp gradients without modifying the geometry parameterization.Source: §6 · p. 30
Detailed mathematical analysis for a priori error estimates and investigation of other PDEs, including wave propagation, remain future work.The authors also identify detailed numerical analysis of GIFT in a boundary element approach as future work.Source: §6 · p. 30

A PHT-splines

PHT-splines are piecewise bicubic polynomial splines on hierarchical T-meshes, offering local refinement and linearly independent blending functions for finite-element approximation.

A PHT-splines

PHT-splines are piecewise bicubic polynomials over hierarchical T-meshes, with simple local refinement and linearly independent blending functions.Unlike T-splines, they are non-rational polynomial splines.Source: A PHT-splines · p. 31
A T-mesh partitions a planar domain into rectangular cells using grid lines parallel to the boundary, and permits T-junctions.Interior vertices are crossing vertices of valency 4 or T-vertices of valency 3.Source: A PHT-splines · p. 31
The spline space consists of piecewise bivariate polynomials of degree (p, q) with continuity orders α and β in the two parametric directions.Source: A PHT-splines · p. 31
For cubic PHT-splines, the dimension is 4(Vb + V+), assigning four basis functions to each boundary or interior crossing vertex.Source: A PHT-splines · p. 31
The initial mesh uses standard bi-cubic tensor-product B-splines, with four basis functions per vertex supported on neighboring grid intervals.The basis functions and their derivatives vanish outside that support.Source: A PHT-splines · p. 31

A.1 Construction of common parametric domain for NURBS geometry and PHT-splines solution

GIFT first constructs a parametric domain for the PHT-spline solution, then solves and locally refines the field space while retaining the NURBS geometry parameterization.

A.1 Construction of common parametric domain for NURBS geometry and PHT-splines solution

Single NURBS patches and multi-patches with quad-mesh parametric domains use the knot-line partition as the PHT-spline solution mesh.Source: §A.1 · p. 32
Other patch layouts require constructing an initial T-mesh from patch connectivity and reparameterizing the patches to form a common parametric domain.The example shifts the parametric domains of adjacent patches while leaving the first patch unchanged.Source: §A.1 · p. 32
A parameter transformation maps each original patch domain to its new parametric domain to obtain a reparameterized surface representation.Source: §A.1 · p. 32
The local-refinement algorithm solves, estimates error, marks cells, subdivides marked cells, rebuilds the basis, and repeats until the error threshold is met.The required derivatives may be evaluated from the original parameterization through a Jacobian transformation, without explicitly deriving the reparameterized surface.Source: §A.1 · p. 33
The solution is represented initially in PHT-spline form on the constructed planar T-mesh, with control variables solved in the model-problem solve.Source: §A.1 · p. 33

A.2 Solution process and local refinement

GIFT refines the numerical solution space locally while leaving the computational-domain parameterization unchanged.

A.2 Solution process and local refinement

After constructing the PHT-spline representation, the control variables are obtained by solving the problem.Source: §A.2 · p. 33
Unlike classical IGA, GIFT performs refinement only on the numerical solution, leaving the domain parameterization unchanged.Source: §A.2 · p. 33
PHT-splines provide natural local refinement, applied on solution sub-patches whose error indicators exceed a marking threshold.Source: §A.2 · p. 33

A.2.1 Residual-based error indicator

The residual-based estimator is expressed through local error indicators on the numerical solution’s parametric T-mesh, matching GIFT’s refinement there rather than on NURBS knot-span sub-patches.

A.2.1 Residual-based error indicator

GIFT refinement requires an error indicator for each cell of the numerical solution’s parametric T-mesh, not an indicator for each NURBS knot-span sub-patch.The refinement is performed on the numerical solution.Source: §A.2.1 · p. 33
The residual-based a posteriori estimate over the parametric domain is rewritten to obtain a local indicator eK for each parametric cell K.The supplied passages do not give the estimator’s full expression.Source: §A.2.1 · pp. 33–34
The physical mapping is x = F(ξ, η), and hK is the circumference of the NURBS-parameterized sub-patch associated with cell K.The estimate also uses a positive constant C.Source: §A.2.1 · p. 33

A.2.2 Marking strategies

The mean-value strategy marks cells whose local error estimate meets or exceeds the mean; a modification first marks a small highest-error fraction when error estimates vary sharply.

A.2.2 Marking strategies

The mean-value strategy marks a parametric cell K for refinement when its local estimate eK is at least the mean estimate across cells.The algorithm takes the parametric T-mesh and cell estimates as input and returns marked cells.Source: §A.2.2 · p. 34
When estimates are highly uneven, the modification first marks the ϵ% of cells with largest errors, then applies mean-value marking to the remaining cells.The authors note that refining only the few cells with extremely large errors can deteriorate local h-refinement performance.Source: §A.2.2 · p. 34

A.2.3 Local refinement

Local refinement updates PHT-spline basis functions on the refined T-mesh while leaving the geometry parameterization unchanged; the refined numerical solution is obtained by solving the PDE again.

A.2.3 Local refinement

Adding a cross vertex refines a cell into four subcells, each supporting the original basis function with 16 Bézier coefficients.The refinement also introduces five new vertices and may introduce new basis vertices.Source: §A.2.3 · p. 34
On refinement, old basis functions are modified and new ones are added; coefficients associated with new basis vertices are reset to zero for old functions.New basis vertices may be crossing vertices or prior-level T-vertices that become basis vertices when neighboring cells gain cross points.Source: §A.2.3 · p. 34
Only the PHT-spline basis on the locally refined T-mesh must be updated; the new control variables are obtained by re-solving the PDE, not by transferring old variables.The refinement applies to the numerical solution.Source: §A.2.3 · p. 34