A stable interface‐enriched formulation for immersed domains with strong enforcement of essential boundary conditions

Abstract

Aims and ScopeThe International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome. Manuscripts should have sufficient original numerical content, and generate new knowledge that is applicable to general classes of engineering problems, and not be limited to applications of existing methods, or propose incremental improvements to existing methods. The journal publishes full-length papers, which should normally be less than 25 journal pages in length, and short communications, which can be at most 8 journal pages. Discussions of papers in print can be published, but two-part papers will not be considered for review.

Smoothed numerical manifold method with physical patch‐based smoothing domains for linear elasticity

“…The standard Lagrange multiplier method caused the boundary locking and oscillation, and some techniques are suggested to overcome it, for example, the enriched formulation 59 . By contrast, the second is to adjust the approximation of the primary variable to let it satisfy the Dirichlet boundary condition exactly, for example, the one proposed by Zheng et al 60 for the element‐free Galerkin method and another approach proposed by van den Boom et al 61 for the discontinuity‐enriched FEM 62,63 . The Nitsche method 58 has received substantial attention recently and has been applied to the meshfree method 64,65 and the partition‐of‐unity methods 66‐71 .…”

A critical view on the use of Non‐Uniform Rational B‐Splines to improve geometry representation in enriched finite element methods

“…However, in this situation, the enriched DOFs are zero, as there is no jump in the gradient of the field in a rigid body displacement of the full element. For more rigorous testing, we place the NURBS interface in an immersed setting, 22 where a material with properties E 1 = 10 and 1 = 0.3 is placed on one side of the interface, and the other side is a void, and the original nodes of the void domain are fixed. This is shown in Figure 7(D)-(I), where we expect, again, to recover three zero-energy modes.…”

“…It has been observed that one factor potentially affecting the accuracy of the computed field gradient is the aspect ratio of the integration elements; 21,22 the creation of enriched nodes too close to existing mesh nodes may result, in some cases, in nonphysical stress concentrations; these are more pronounced near material interfaces 46,47 rather than along Dirichlet boundaries 22 or traction-free cracks. 21 The occurrence of tangent discontinuities or close-to-coincident enriched TA B L E 2 Normalized K I and K II , obtained analytically, with NURBS-based DE-FEM, and with standard DE-FEM…”

“…Figure 17 illustrates this immersed problem schematically. On the boundary Γ i , a radial displacement u(r, ) = [−0.5r, 0] ⊺ is prescribed; following the immersed Dirichlet BCs described in van den Boom et al 22 that mimics a uniform compression of the duck that leads to a uniform state of stress inside it. All original mesh nodes in the void region are fixed.…”

“…Since its inception, DE-FEM has been extended to the analysis of 3D problems 21 and immerse boundary (fictitious domain) problems. 22 The latter work showed that the enriched formulation is capable of recovering smooth traction profiles at Dirichlet boundaries, something that is not possible with X/GFEM even with stabilization. 23 While the vast majority of literature on X/GFEM and interface-enriched FEM approximates geometries piecewise linearly, higher-order piecewise polynomial parametrizations have been proposed to approximate curved discontinuities.…”