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While the literature on numerical methods (e.g. finite elements and, to a certain extent, virtual elements) concentrates on convex elements, there is a need to probe beyond this limiting constraint so that the field can be further explored and developed. Thus, in this paper, we employ the virtual element method for non-convex discretizations of elastodynamic problems in 2D and 3D using an explicit time integration scheme. In the formulation, a diagonal matrix-based stabilization scheme is proposed to improve performance and accuracy. To address efficiency, a critical time step is approximated and verified using the maximum eigenvalue of the local (rather than global) system. The computational results demonstrate that the virtual element method is able to consistently handle general nonconvex elements and even non-simply connected elements, which can lead to convenient modeling of arbitrarily-shaped inclusions in composites.

Starting from a highly continuous isogeometric analysis discretization, we introduce hyperplanes that partition the domain into subdomains and reduce the continuity of the discretization spaces at these hyperplanes. As the continuity is reduced, the number of degrees of freedom in the system grows. The resulting discretization spaces are finer than standard maximal continuity IGA spaces. Despite the increase in the number of degrees of freedom, these finer spaces deliver simulation results faster with direct solvers than both traditional finite element and isogeometric analysis for meshes with a fixed number of elements. In this work, we analyze the impact of continuity reduction on the number of Floating Point Operations (FLOPs) and computational times required to solve fluid flow and electromagnetic problems with structured meshes and uniform polynomial orders. Theoretical estimates show that for sufficiently large grids, an optimal continuity reduction decreases the computational cost by a factor of O ( p 2 ) . Numerical results confirm these theoretical estimates. In a 2D mesh with one million elements and polynomial order equal to five, the discretization including an optimal continuity pattern allows to solve the vector electric field, the scalar magnetic field, and the fluid flow problems an order of magnitude faster than when using a highly continuous IGA discretization. 3D numerical results exhibit more moderate savings due to the limited mesh sizes considered in this work.

We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preserving the optimal convergence rates, as confirmed by both the theoretical estimates and the numerical results.

In this paper, we propose a diffuse interface model and finite element approximation for two-phase magnetohydrodynamic (MHD) flows with different viscosities and electric conductivities. An energy stable scheme, which is based on the finite element method for the spatial discretization and first order semi-implicit scheme combined with convex splitting method for the temporal discretization, is proposed to solve this new model. The numerical scheme is proved to be mass-conservative and energy law preserving. By Leray–Schauder fixed point theorem, the existence of solutions to the numerical scheme is shown. The uniqueness of the numerical solutions is obtained. Utilizing the stability of the numerical scheme and the compactness method, the existence of the weak solutions to the two-phase MHD model is established as well. Furthermore, given more regularity on the weak solution, the convergence of the numerical scheme is derived. Finally, numerical experiments are provided to verify the theoretical results and validate the proposed model.

Two triangular G 1 -conforming elements, based on the triangular Gregory patch, suitable for the analysis of the Kirchhoff plate model are presented. Both have cubic normal derivative along the sides, so that they can be effectively used in combination with generalized Hermitian elements. The Gregory patch consists in a rational enhancement of the base-polynomial spaces useful to design G 1 -conforming elements on general C 0 -conforming unstructured meshes. Because of the presence of the rational functions, the second derivatives at the corners of the element present finite discontinuities, that prevent the elements from passing the bending patch test. The discontinuities are removed using a constrained version of the Gregory patch, with Lagrange multipliers. In this way, the rational conforming space collapses into a conforming rearrangement of the original polynomial interpolant spaces. The proposed formulation design elements that pass the bending patch test and present optimal rate of convergence on general unstructured meshes.

This paper presents a new high-order mixed scheme of polygonal finite element method (PFEM) for incompressible flow computation. The system of polygonal basis functions with bubble-enriched functions is applied to represent the fluid velocity field, while the pressure field is approximated by the set of polygonal basis functions. The new element is named as MINIPe. From a practical point of view, the present element naturally satisfies the well-known inf–sup condition. The numerical tests of the incompressible fluid flow problems governed by Stokes equation systems are presented to verify the performance quality of proposed method with respect to the stability and convergence rate.

This paper proposes a new multiscale topology optimization method for the concurrent design of multiphase composite structures under a certain range of excitation frequencies. Distinguished from the existed studies, a general concurrent design formulation for the dynamic composite structures with more than two material phases is developed. The macrostructureand its microstructures with multiple material phases are optimized simultaneously. The integral of the dynamic compliances over an interval of frequencies is formulated as the optimization objective, so as to minimize the frequency response within the concerned excitation range. The effective properties of the multiphase microstructures are evaluated by using the numerical homogenization method, which actually serves as a link to bridge the macro and micro finite element analyses. Furthermore, to describe the boundaries of multiple material phases for the microstructure, a parametric color level set method (PCLSM) is developed by using an efficient interpolation scheme. In this way, L level set functions can represent at most 2 L material phases without any overlaps. Moreover, these “color” level sets are updated by directly using the well-established gradient-based algorithm, which can greatly facilitate the proposed method to solve the multi-material optimizations with multiple design constraints. Several 2D and 3D numerical examples are used to demonstrate the effectiveness of the proposed method in the concurrent design of the dynamic composite structures under the excitation frequency ranges.

The computational modelling of the heart motion within a cardiac cycle is an extremely challenging problem due to (a) the complex multi-scale interaction that takes place between the electrophysiology and electrochemistry at cellular level and the macro-scale response of the heart muscle, and (b) the large deformations and the strongly anisotropic and quasi-incompressible behaviour of the myocardium. These pose an extreme challenge to the scalability of electro-mechanical solvers due to the size and conditioning of the system of equations required to obtain accurate solutions, both in terms of wall deformation and transmembrane potential propagation. In the search towards an efficient modelling of electro-activation, this paper presents a coupled electromechanical computational framework whereby, first, we explore the use of an efficient stabilised low order tetrahedral Finite Element methodology and compare it against a very accurate super enhanced mixed formulation previously introduced by the authors in Garcia-Blanco et al. (2019) and, second, we exploit the use of tailor-made staggered and staggered linearised solvers in order to assess their feasibility against a fully monolithic approach. Through a comprehensive set of examples, culminating in a realistic ventricular geometry, we aim to put forward some suggestions regarding the level of discretisation and coupling required to ensure sufficiently reliable results yet with an affordable computational time.

This paper presents an approach to the automatic enrichment of finite elements in the vicinity of a stress singularity. The enrichment consists of semi-analytical singular modes constructed using the Scaled Boundary Finite Element Method (SBFEM). In contrast to analytical methods, the SBFEM provides modes for inhomogeneous and anisotropic materials without additional effort. The finite element basis can be of arbitrary order and remains unaltered by the enrichment. The approach requires enrichment in only one layer of elements around a node. Due to the compatibility of SBFEM with FEM, there is no need for transitional elements, and there are no parasitic terms. The approach is tested for several benchmark problems. The stress intensity factors are computed based on techniques inspired by the SBFEM. The proposed procedure is compared to a standard finite element implementation and shows a significant improvement in the error of the displacement field for problems involving singular stresses.

The aim of this paper is to accurately solve short wave scattering problems governed by the Helmholtz equation using the Bernstein–Bezier Finite Element method (BBFEM), combined with a conformal perfectly matched layer (PML). Enhanced PMLs, where curved geometries are represented by means of the blending map method of Gordon and Hall, are numerically investigated. In particular, the performance of radial and elliptical shaped PMLs, with a parabolic absorption function, are assessed and compared in terms of accuracy against second order Bayliss–Gunzberger–Turkel (BGT 2 ) based local absorbing boundary conditions. Numerical results dealing with problems of Hankel source radiation and wave scattering by a rigid cylinder show that the radial shaped PML, with the h and p versions of BBFEM, enables the recovery of the predicted algebraic and exponential convergence rates of a high order finite element method (FEM). Furthermore, radial shaped BGT 2 and PML have a similar performance, as long as the wave is not sufficiently well resolved. But, BGT 2 performs poorly as the wave resolution increases. Additionally, the effect of harmonics of higher modes on accuracy is examined. The study reveals that the PML outperforms BGT 2 for almost all propagating modes. However, a similar performance is achieved with both methods either with higher modes or a low wave resolution. Results from a multiple scattering benchmark problem provide evidence of the good performance of the proposed PMLs and the benefit of elliptical shaped PMLs in reducing significantly the size of the computational domain, without altering accuracy. The choice of the PML parameters ensuring optimal performance is also discussed.

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) , and L 2 ( Ω ) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nedelec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy’s problems based on compatible pairs of approximations in { H ( div , Ω ) , L 2 ( Ω ) } for such tetrahedral–hexahedral–prismatic–pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.

In this work, we propose an adaptive phase field method (PFM) to simulate quasi-static brittle fracture problems. The phase field equations are solved using the scaled boundary finite element method (SBFEM). The adaptive refinement strategy is based on an error indicator evaluated directly from the solutions of the SBFEM without any need for stress recovery techniques. Quadtree meshes are adapted to perform mesh refinement. The polygons with hanging nodes in the quadtree decomposition are treated as n − sided polygons within the framework of the SBFEM and do not require any special treatment in contrast to the conventional finite element method. Several benchmark problems are used to demonstrate the robustness and the efficacy of the proposed technique. The adaptive refinement strategy reduces the mesh burden when adopting the PFM to model fracture. Numerical results show an improvement in the computational efficiency in terms of the number of elements required in the standard PFM without compromising the accuracy of the solution.

Peridynamics is a nonlocal reformulation of solid mechanics which can reproduce the solution of classical continuum mechanics as its horizon approaches zero. Taking this idea into consideration, we proposed a new coupling approach between meshfree peridynamics and finite element method. An intermediate variable horizon element called “morakkab” is introduced to transfer force and momentum between peridynamic and FEM regions. In the original peridynamic formulation, artificial ghost forces will arise if the horizon varies with position; The creation of these artifacts is avoided by utilizing a modified peridynamic equation of motion based on the concept of co-family. The accuracy of the method is illustrated in 2D benchmark tests. Using the modified equation of motion, compared to the original peridynamic formulation, dramatically decreases the relative errors with respect to the analytical solution.

The paper presents a new two-scale methodology for modelling of damage in ductile heterogeneous materials employing homogenization approach over the two microstructural models. The first microstructural model comprises elastoplastic behaviour without softening, while only microstructural softening is homogenized in the second model. The scale transition of the state variables is based on the first-order computational homogenization scheme. The microstructural damage response is governed by the nonlocal variable embedded into the gradient elastoplastic formulation using the implicit localization approach. The averaging computation over the microstructural softening zones is driven by the damage variable and the local equivalent strain. Therein discretization is performed by using the mixed finite element formulation including the nonlocal equivalent plastic strain interpolation, while the standard displacement based finite elements, employing the simple constitutive relation with damage variable, are used at the macrolevel. The capability of modelling material damage responses at both the micro and macroscale is demonstrated by the benchmark examples.

The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to special techniques to truncate infinite domains. The isogeometric boundary element method (IGABEM) is a recent progress in the category of boundary element approaches, which is inspired by the concept of isogeometric analysis (IGA) and employs the spline functions of CAD as basis functions to discretize unknown physical fields. As a boundary representation approach, IGABEM is naturally compatible with CAD and thus can directly perform numerical analysis on CAD models, avoiding the cumbersome meshing procedure in conventional FEM/BEM and eliminating the difficulty of volume parameterization in isogeometric finite element methods. The advantage of tight integration of CAD and numerical analysis in IGABEM renders it particularly attractive in the application of structural shape optimization because (1) the geometry and the analysis can be interacted, (2) remeshing with shape morphing can be avoided, and (3) an optimized solution returns a CAD geometry directly without postprocessing steps. In the present paper, we apply the IGABEM to structural shape optimization of three dimensional exterior acoustic problems, fully exploiting the strength of IGABEM in addressing infinite domain problems and integrating CAD and numerical analysis. We employ the Burton–Miller formulation to overcome fictitious frequency problems, in which hyper-singular integrals are evaluated explicitly. The gradient-based optimizer is adopted and shape sensitivity analysis is conducted with implicit differentiation methods. The design variables are set to be the positions of control points which directly determine the shape of structures. Finally, numerical examples are provided to verify the algorithm.

This paper applies high-order discontinuous Galerkin finite element solutions of the shallow water equations to realistic coastal hydrodynamics problems. The goal is to obtain results on par with the accuracy of current low-order models, while significantly surpassing their performance. Accomplishing this requires developing mesh discretizations with boundary and parameter field representations that are consistent with the high-order accuracy of the numerical methods. These developments allow coarse-mesh, high-order, solutions to provide comparable accuracy to today’s state-of-the-art high-resolution, low-order coastal models at reduced computational expense.

A 3D fluid–structure coupling between Smoothed Particle Hydrodynamics (SPH) and Finite Element (FE) methods is proposed in this paper, with its application to complex tire hydroplaning simulations on rough ground. The purpose of this work is to analyze the SPH–FE coupling capabilities for modeling efficiently such a complex phenomenon. On the fluid side, the SPH method is able to handle the three complex interfaces of the hydroplaning phenomenon: free-surface, ground/fluid and fluid/tire interfaces. On the solid side, the FE method is used for its ability to treat tire–ground contact. A new algorithm dedicated to such SPH–FE coupling strategies is proposed to optimize the computational efficiency through the use of differed time steps between fluid and solid solvers. This way, the number of calls to the FE solver is minimized while maintaining the accuracy and stability of the coupling. The ratio between these respective time steps relies on a control procedure based on pressure loading. The present 3D SPH–FE model is first validated with different academic test cases and experimental data before considering the complex problem of the 3D hydroplaning simulations. Hydroplaning simulations are performed and analyzed on 3D configurations involving both smooth and rough grounds.

Composite shells with complex geometry are widely used in aerospace structures. Due to the complexity of geometry and curvilinear fiber path, the analysis and optimization based on finite element analysis (FEA) for complex variable-stiffness (VS) shells is extremely time-consuming. By comparing with FEA, isogeometric analysis (IGA) exhibits higher prediction efficiency of buckling load. In this work, the formula of geometric stiffness matrix for complex VS shells is derived for the first time based on degenerated shell method using IGA, which is the basis of performing linear buckling analysis. Then, a new variable curvature quasi-linear function (VCQLF) to describe curvilinear fiber path is proposed, which can further expand the design space of VS shells. After that, two frameworks for shape optimization of complex shells are put forward and then compared, and it is found that the one based on LOFT function can provide representative control variables of shape and effectively reduces the number of design variables for complex shells. Finally, a novel collaborative optimization framework of fiber path and shell shape using IGA is established. By comparison of traditional methods, it is demonstrated that the proposed framework can greatly improve the efficiency of optimization and fully explore the buckling load of complex VS shells.

A block coordinate descent algorithm within a discontinuous Galerkin finite element formulation is proposed for both explicit and implicit energy minimization in solid bodies in which rigid-cohesive fractures initiate and propagate. The crack opening dependent Barenblatt type surface cohesive energy function is non-differentiable at the origin. In each iteration or each explicit update, the algorithm minimizes the potential with respect to each crack opening displacement unknown and with respect to the block of deformation unknowns, sequentially. This decomposes minimization of the full non-differentiable problem into a number of “small” non-differentiable sub-problems that must be solved locally at the inter-element boundaries of the finite element mesh and an “easy” differentiable sub-problem that characterizes global equilibrium. As a result, non-differentiability is effectively treated locally and the algorithm can be easily incorporated into standard finite element codes. On the basis of a convexity analysis of the proposed non-differentiable energy functional, we obtain a minimum cohesive process zone resolution criterion, known empirically in the previous literature as a requirement for capturing the amount of dissipated fracture energy correctly. The method is free of any regularization parameters and preserves the discrete nature of fracture without introducing a stress discontinuity at initiation of decohesion. Robustness of the method is shown through several numerical examples of fragmentation and branching and through comparisons with existing numerical and experimental results.

We derive a thermodynamically-consistent, three-dimensional, rate form-based finite-deformation constitutive theory and computational approach for damage & fracture in nonlinear viscoelastic materials. The key ingredient that allows us to develop a physical criterion for fracture is the use of a Gibbs potential-based multinetwork formulation of viscoelasticity. An approach that is based on the use of the criticality of the averaged Gibbs potential over a fracture process zone is used for the initiation & propagation of cracks in a body. The rate form-based constitutive theory and fracture criterion are implemented into the Abaqus (2018) finite-element program through a user-material subroutine interface. Crack propagation is modeled as the failure of elements leading to loss of mechanical resistance to deformation. The tearing & fracture response of a viscoelastic material using a notch-in-plate sample deformed in simple tension is also simulated. By comparing the simulation results for a local damage criterion versus a truly nonlocal damage criterion, we show how the spread of Gibbs free energy around the crack tip influences the nature of the crack growth. As expected, the material fracture response described by a local damage criterion exhibits pathological mesh dependence whereas the response obtained using the nonlocal damage criterion is mesh objective regardless of mesh density, element type and orientation. This work presents two major computational advances: (1) the conversion of the rate form-based finite-deformation constitutive theory into an objective time-integration procedure is straightforward because the material time derivative-based constitutive equations are frame-invariant, and (2) the physically-observed crack initiation & propagation process in solids can be accurately & robustly simulated using simple numerical techniques such as the element failure method if a truly nonlocal fracture criterion is utilized. Finally, we benchmark results obtained from our computational framework and numerical simulations for modeling crack initiation & propagation to physical experimental data of mixed-mode crack propagation in a viscoelastic material.

In order to exclude spurious failure in compression the consideration of tension compression asymmetry (TCA) in damage models is of great interest in research as well as in industry. This paper presents an efficient and unifying algorithmic procedure for the incorporation of TCA into damage models. With the presented approach the implementation effort is drastically reduced since the equations and subroutines of the model without TCA can be reused. The strategy is first introduced and illustrated for an isotropic damage model. Afterwards, the procedure is applied to a gradient-extended anisotropic damage model with a second order damage tensor recently developed by the authors. For this model studies at integration point level as well as structural examples are presented. Two different aspects are included in the presented TCA approach and are demonstrated with the results of the simulations: (i) different damage evolution in tension and compression and (ii) stiffness recovery (crack closure) under compression. Due to the utilized gradient-extended formulation, being very efficient since only one additional degree of freedom is introduced, the finite element computations are shown to deliver mesh-objective results.

This study presents a novel multi-scale computational method to analyze the dynamic thermo-mechanical performance of heterogeneous shell structures with orthogonal periodic configurations. The heterogeneities of heterogeneous shell structures are taken into account by periodic layouts of unit cells on the microscale in orthogonal curvilinear coordinate system. The new second-order two-scale approximate solutions for these multi-scale problems are constructed based on the multi-scale asymptotic analysis. Furthermore, the error estimates for the second-order two-scale (SOTS) solutions are obtained under some hypotheses. And then, a novel SOTS numerical algorithm based on finite element method (FEM), finite difference method (FDM) and decoupling method is brought forward in detail. Finally, some numerical examples are presented to verify the feasibility and validity of our multi-scale computational method. They also demonstrate that our multi-scale computational method can accurately capture the micro-scale dynamic thermo-mechanical responses in heterogeneous block structure, plate, cylindrical and doubly-curved shallow shells. In this paper, a unified multi-scale computational framework is established for dynamic thermo-mechanical problems of heterogeneous materials and structures with orthogonal periodic configurations. The asymptotic homogenization theory in Cartesian coordinate system and cylindrical coordinate system can be directly obtained based on the results in this paper.

A new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Laplace equation on irregular domains with the Dirichlet boundary conditions has been developed. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used for 3-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances ( 0 . 1 h − 1 0 − 9 h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fifth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.

In the past the Vertex Morphing method has mainly been used in the context of optimization of thin walled shell structures or fluid mechanical problems. In this paper an extension of this parameter free shape optimization method to 3-dimensional solid structures based on the free Finite Element code CalculiX is presented. Key element of the whole process is an efficient adjoint sensitivity analysis that can handle many design variables providing gradient information for the response functions of interest, e.g. stresses and mass. Typically, parameter free shape optimal designs result in non-smooth geometries and suffer from mesh dependencies. To avoid these phenomena an intermediate filter operation in combination with several post processing steps of the raw sensitivities is added. In the optimization, the filter size is also used as an additional design parameter controlling the curvature of the optimal shape. In many engineering optimization applications constraints have to be considered. These restrictions are included in the optimization problem by the use of Rosen’s gradient projection method. For the successive shape update a pseudo linear-elastic static Finite Element analysis with multiple point constraints and a varying Young’s modulus is performed in every design iteration . Finally, the capabilities of the extended Vertex Morphing method are demonstrated for an academic example and a geometrically complex part from aerospace industry.

In terms of the three-field (u-w-p ) formulation of Biot’s theory of saturated porous media with incompressible solid and fluid phases, the numerical manifold method (NMM) models are developed to analyze the fully dynamic consolidation of fractured porous media in this study. The same approximation to fluid velocity and skeleton displacement is constructed which is capable of modeling both incompressible and compressible deformation, while two types of approximations to pore pressure field are established. Since the inertial effect of fluid is not neglected, the proposed model can fully capture the dynamic behavior of porous media, especially under the impact or high-frequency loading condition and, accordingly, exhibits apparent superiority in predicting transient and wave propagation responses of cracked porous media. Moreover, low order interpolation functions for primal variables and the most flexible three-node triangular finite element mesh are used, which both are difficult to implement using other partition of unity (PU) based numerical methods in terms of Biot’s two-field formulation. Also, the discontinuities can be modeled more naturally in the NMM framework in comparison with XFEM or PNM . Meanwhile, an augmented Lagrange multiplier method for stick–slip contact model is first incorporated into fully-dynamic three-field Biot’ formulation. In addition, a mass lumping technique within NMM framework, which turns out to be a unique advantage of NMM over other numerical methods, is employed to suppress unphysical oscillations and increase computational efficiency. Energy balance condition is employed to evaluate the stability and accuracy of the time integration scheme. The robustness and versatility of the proposed models are manifested with several typical examples.

The ability to make decisions based on quantities of interest that depend on variables inferred from measurement finds application in different fields of mechanics and physics. The evaluation of the inferred variables, and hence the quantities of interest, from the measurement typically requires the solution of an inverse problem . For example, in medical imaging the elastic heterogeneity of a tumor and its nonlinear elastic response can be used to distinguish benign tumors from their malignant counterparts. These images of linear and nonlinear elastic parameters of tissue are typically obtained by using a measured displacement field and solving a complex inverse elasticity problem . In this paper we consider circumventing the solution of the inverse problem by using measured displacements as input to a deep convolutional neural network (CNN) and training it to classify tumors on the basis of their elastic heterogeneity and nonlinearity. For a simple, 5-layer CNN trained with 8,000 samples for heterogeneity, and a 4-layer CNN trained with 4,000 samples for nonlinear elasticity we report classification accuracies in the range of 99 . 7 % − 99 . 9 % . The training and testing data are both obtained from the forward solution of finite element models of samples. We also analyze the weights of the trained model to understand the process through which the network extracts features of elastic moduli from the input displacement images. Finally, we apply the nonlinear elasticity classifier , which is trained entirely using simulated data, to displacement images obtained from ten patients with breast lesions and note that it correctly classifies eight out of ten cases. This application illustrates how data from physics-based models can be used in improving the performance of a data-driven algorithm in data-sparse scenarios.

There is growing interest in high fidelity simulations of buoyancy-driven flows in indoor environments. This is driven to a large extent by the push to utilize naturally occurring flows to reduce energy consumption in the built environment. Naturally occurring (buoyancy-driven) flows exhibit spatiotemporal thermal fluctuations, and accurately capturing these thermal variations is essential to calculate energy usage. In this work, we deploy a residual-based variational multiscale (VMS) finite element large-eddy simulation model for accurately modeling buoyancy-driven flows in enclosed environments. We use the canonical example of the Rayleigh–Benard convection problem in 2D and 3D to verify and validate the VMS model. We show good comparison with benchmark numerical and experimental results across seven orders of magnitude variation in Rayleigh numbers ( R a ∼ 1 0 3 to 1 0 10 ), covering laminar, transition, and turbulent regimes without any extra treatments. We additionally employ weak enforcement of Dirichlet boundary conditions for both velocity and temperature, and show that comparable results can be produced with much coarser meshes. This confirms that the VMS framework with weak imposition of Dirichlet boundary conditions is a computationally efficient approach to model buoyancy-driven flow physics in complex indoor environments.

Shear localization is a damage mechanism characterized by the concentration of plastic deformation within thin bands of materials, often observed as a precursor to failure in metals. In this paper, we present a three-dimensional explicit finite element formulation with embedded weak discontinuities for the treatment of shear localization under dynamic loading conditions. In this formulation, the onset of localization is detected via a material stability analysis suitable for rate-sensitive materials, and the continuity of propagating shear bands is ensured using a global tracking strategy involving a heat-conduction type boundary value problem, solved for a scalar level set function over the global domain of the problem. In addition, we propose a modified quadrature rule that is used to compute the contribution of an individual element to the global finite element arrays, taking into account the position of the embedded shear band within that element. The mechanical threshold stress (MTS) model, modified to account for dynamic recrystallization, is adopted as the constitutive model for rate- and temperature-sensitive metallic materials, and the constitutive equations are formulated in a corotational setting. The algorithmic implementation of the integrated finite element framework, including the global tracking strategy, is described in detail. Subsequently, a simple example problem involving shear band formation under uni-axial tension is presented, to illustrate the advantages of the proposed computational framework over conventional finite element techniques, in terms of convergence and mesh insensitivity of their numerical results. In addition, to demonstrate the capabilities of the proposed computational framework in more realistic problems, numerical results are compared to data from split-Hopkinson pressure bar dynamic experiments, conducted on two stainless steel samples with different geometric designs. In each problem presented, we study the effect of the mesh size on the numerical solution, including global results such as the load–displacement response, and local quantities such as material state variables at individual quadrature points. These numerical studies show that the geometry and total volume of the fully-developed embedded shear band are independent of the mesh size.

In this paper, we present an isogeometric analysis (IGA) for phase-field models of three different yet closely related classes of partial differential equations (PDEs): geometric PDEs, high-order PDEs on stationary surfaces, and high-order PDEs on evolving surfaces; the latter can be a coupling of the former two classes. In the context of geometric PDEs, we consider mean curvature flow and Willmore flow problems and their corresponding phase-field approximations which yield second-order and fourth-order nonlinear parabolic PDEs. Through some numerical examples, we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions. Moreover, we study numerically the convergence of these phase-field approximations to the sharp interface solutions . As for the high-order PDEs on stationary surfaces, we consider a model problem which is the Cahn–Hilliard equation on a unit sphere, where the surface is modeled using a diffuse-interface approach. Finally, as a model problem for high-order PDEs on evolving surfaces, we consider a phase-field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface. The model consists of two fourth-order nonlinear PDEs which their direct finite element formulation in a Galerkin framework necessitates smooth basis functions with at least global C 1 continuity; a condition that can be easily satisfied using spline bases in IGA. We solve the coupled equations both in two dimensions, where the vesicle is a curve, and in three dimensions, where the vesicle is a surface. The simulation results agree with the numerical and experimental results from the literature.

Composite structural failure usually entails complicated interactions between intralaminar and interlaminar damage. In this paper, a novel discontinuous solid-shell finite element combined with a 3D enriched cohesive element is formulated for the analysis of matrix cracking and delamination in multilayered composites. The proposed formulations are computationally attractive in modeling thin shell structures while retaining the advantages of solid elements. The bending performance was improved by using the enhanced assumed strain method and the assumed natural strain method, which alleviates potential locking problems. A linear elastic orthotropic law was used for layered shells and cracks therein were modeled with a cohesive zone model. A discrete crack method with floating nodes enriching the developed elements enables a mesh-independent description of ply cracks and their interaction with interface cracks. The applicability of the current formulation was verified with numerical examples including large deformations of thin shell problems and delamination growth in post-buckled laminates. Finally, the explicit and totally discrete modeling of matrix and delamination cracks in low velocity impact of cross-ply and angle-ply laminates was demonstrated. Numerical predictions of structural responses and damage patterns agree well with experimental results.

In this paper, we present a computational homogenization framework to model coupled transient heat conduction with heat generation and chemical kinetics in solids. The proposed method considers that both macro and microscales are under transient heat conduction, being the chemical kinetics (which gives rise to internal heat generation source in the heat conduction problem) defined either in some of the constituents of the microscale or even in the whole microscale. The numerical solution is based on a nested solution strategy, in which the finite element method is used for solving both the macro and the microscale problems, configuring a FE 2 scheme. By solving the coupled problem of transient heat conduction with internal heat generation and chemical kinetics defined in a finite representative volume element , we extract the effective thermal properties and chemical kinetics contribution of the composite and employ them to solve the transient macroscale heat conduction problem (also with internal heat generation). This novel numerical framework is employed in the prediction of the State Of Cure (SOC) in filled elastomers . Numerical solutions of some in-plane heat conduction problems are presented in order to assess the proposed numerical strategy, showing that the multiscale model developed is capable of numerically determining transient non-homogeneous maps of the SOC at the microscale.

*The work of Shaw was supported in part by the Engineering and Physical Sciences research council EP/H011072/1. This support is gratefully acknowledged.

Generalized continuum theories become necessary for many scenarios like strain localization phenomena or size effects. In this work, we consider a class of higher order continua, namely micromorphic continua, where the kinematics is enhanced by means of a microstructure undergoing an affine micro deformation. The increasing number of the degrees of freedom in such a theory clearly motivates an application of adaptive methods. For linear elastic micromorphic problems, we have shown the consistency of the resulted dual problem, ensuring an optimal convergence order, in Ju and Mahnken (2017). In this work, we extend the framework of goal-oriented adaptivity to time-dependent problems, i.e. plasticity problems, where a backwards-in-time dual problem is crucial to account for the error accumulation over time caused by both error generation and error transport. Our focus is limited to an adaptive control of spatial discretization errors of the finite element method (FEM), while the temporal discretization errors are not considered for simplicity. Based on duality techniques, we derive exact error representations. For practice, four computable error estimators are proposed, where two different ways to obtain enhanced solutions are considered, and where additionally an approximate forwards-in-time dual problem neglecting error transport is introduced. By means of certain localization techniques, these estimators are used to drive an adaptive mesh refinement algorithm. Their effectiveness is shown by several numerical examples based on a prototype model.

This paper proposes a numerical model for the simulation of fracture propagation in brittle materials based on the theory of configurational forces. Configurational forces are adopted as the driving force for fracture in association with a tensile stretch criterion to identify crack tips of propagating cracks . Fracture surfaces are approximated by zero-thickness interface elements embedded into the finite element mesh between the bulk elements in the entire domain. In order to eliminate the mesh bias in regards to the fracture pattern, a new fracture segment is allowed to grow along several interface elements at once. The minimal fracture surface is found using an algorithm based on Graph Theory. The solution strategy is explicit, and the fracture path using this algorithm is identified in a post-processing step . The proposed configurational interface model (CIM) for fracture is validated by the analytical solution for a mode I benchmark and by comparisons with results from a variety of laboratory experiments on cylindrical granite specimens containing one and two initial cracks at different angles, which are characterized by complex curved and partially irregular crack patterns.

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide . The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient . We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.

In this work we review a unified formulation for spectral fuzzy, spectral stochastic, and spectral fuzzy–stochastic FEM. We propose some modifications for the fuzzy and fuzzy–stochastic FEM involving local bases in the non-deterministic dimensions and the incorporation of extended FEM into the non-deterministic FEM. These modifications were previously used for stochastic FEM only. We discuss advantages of the proposed techniques for problems with uncertainties in the geometry and demonstrate their application to computational homogenization of heterogeneous materials with geometrical uncertainties in the microstructure. We address also some theoretical aspects of fuzzy FEM, that have not been tackled in the literature, namely the inner product of fuzzy variables. We address also another important problem, which becomes often the target of criticism of fuzzy approach. The membership function of the fuzzy input has often some degree of arbitrariness: it is known only approximately or it is constructed based on some assumptions. Here we propose a fuzzy FEM representation, which requires only the modal value and support of the fuzzy input parameters and is completely independent of the membership function’s shape. We also discuss the case of imprecise probabilities which allows for severe dimension reduction of the non-deterministic problem if the accurate spectral simulation technique is used. Application of dimension reduction is also demonstrated on the example of computational homogenization.

In this paper we consider fully mass-conservative numerical schemes for the simulation of incompressible and immiscible two-phase flow in porous media with capillary pressure. Compared with two kinds of conventional IMplicit Pressure Explicit Saturation (IMPES) schemes, the new schemes deserve a merit that the conservation of mass of both phases can be obtained. The total conservation equation is obtained by the summation of the discretized conservation equation for each phase. This approach is quite different from the conventional IMPES schemes. We present two kinds of fully mass-conservative IMPES schemes to solve the coupled systems for pressure, auxiliary velocity and saturation of each phase. The upwind mixed finite element methods are used to solve the pressure–velocity systems which can be decoupled, and the problems in the decoupled systems can be proved to be well-posed. Moreover, the new schemes are unbiased and the saturation of each phase can be proved to be bounds-preserving if the time step size is smaller than a certain value. The new schemes can also be applied to approximate the incompressible and immiscible two-phase flow in heterogeneous porous media with different capillarity pressures. Several interesting examples of incompressible and immiscible two-phase flow in porous media are presented to demonstrate the robustness of the new algorithms.

It is well known that Galerkin finite element methods suffer from pollution error when solving wave problems. To reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave-like functions so that the exact wavenumber is incorporated into the finite element approximation space. Solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied. Recently, a plane wave enriched isogeometric analysis was introduced for acoustic problems. However, it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other. In this work, we show that isogeometric analysis outperforms finite element method in dealing with pollution. We observe similar behaviour when both the methods are enriched with plane waves. Using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions. However, the latter still leads to smaller errors using similar number of degrees of freedom. In conclusion, we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re-meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour.

The Bernstein polynomials have been known for over a century and are widely used in the spline literature, computer aided geometric design, and computer graphics. However, the realization that the Bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development. For instance, it is known (Ainsworth et al., 2011) that the Bernstein basis can be exploited to compute all of the entries in the load vector in O ( p 3 ) operations even in the case of non-linear problems on curvilinear elements for a degree p approximation. Moreover, the element matrices can be assembled in O ( 1 ) operations per entry. We show that properties of the Bernstein polynomials can also be exploited to obtain O ( p 3 ) complexity procedures for all of the main components needed to implement a high order finite element code including: computation of the residuals needed for an iterative solution method; evaluating the action of a preconditioner for the global mass matrices; and, visualization and post-processing of the resulting finite element approximations. The construction of a preconditioner for the mass matrix whose condition number does not degenerate with the order p , at a cost of O ( p 3 ) operations, is one of the main contributions of the present work. The preconditioner is based on an abstract Additive Schwarz Method recently developed by the authors. The preconditioner can be implemented at a cost of O ( p 3 ) operations by exploiting properties of the Bernstein polynomials. In particular, we present an algorithm which allows one to invert the interior block of the element mass matrix in O ( p 3 ) operations. Numerical examples are provided to illustrate the applicability of the Bernstein basis to challenging non-linear reaction–diffusion problems, non-linear wave propagation of solitons and to robust approximation of problems exhibiting boundary layers.

In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.

At present, the most widely used mid-frequency method, i.e., the hybrid finite element-statistical energy analysis method (FE/SEA), still holds potential flaws, e.g., low accuracy and ignoring uncertain factors. In order to eliminate these problems, a generalized smoothing finite element statistical energy analysis method (SFE/SEA) is first proposed in this work, which is constructed based on the edge-based gradient smoothing technique, thus reducing the dispersion error in the deterministic calculations. Second, on the basis of the proposed SFE/SEA, the parametric uncertainty models will be introduced, combining the second-order random interval perturbation method and the random interval moment method, etc.; thus, a mid-frequency analysis method (SHPSFEM/SEA) considering both parametric uncertainties (both the random and interval variables) and nonparametric uncertainties is proposed, to effectively analyze the probability and interval characteristics of the random structural–acoustic coupled systems. Moreover, for the interval parameters with wide variation range, the subinterval perturbation method (S-SHPSFEM/SEA) is introduced by dividing the large interval parameters into small interval parameters. The proposed methods are compared with the Monte Carlo simulations of the hybrid FE/SEA model. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.

In this study, a varying/adapting cohesive approach is developed to model crack-healing procedure in reinforced shape memory composites. To this aim, the stress–strain behavior of SMA fiber in thermo-mechanical condition is converted into a traction–separation response between crack faces. The extended finite element formulation is employed to model crack in reinforced composites and to avoid remeshing while maintaining the required accuracy of results. Combination of XFEM to model arbitrary crack and traction–separation formulation enables the model to simulate mixed-mode crack propagation and crack-healing with different fiber orientations. Examining the results shows that the proposed model maintains the precision of direct modeling of SMA fibers while decreasing the computational costs and complexity of modeling significantly.

Accurate and efficient modeling of microstructural interaction and evolution for prediction of the macroscopic behavior of materials is important for material design and manufacturing process control . This paper approaches this challenge with a reduced-order method called self-consistent clustering analysis (SCA). It is reformulated for general elasto-viscoplastic materials under large deformation . The accuracy and efficiency for predicting overall mechanical response of polycrystalline materials is demonstrated with a comparison to traditional full-field solution methods such as finite element analysis and the fast Fourier transform . It is shown that the reduced-order method enables fast prediction of microstructure–property relationships with quantified variation. The utility of the method is demonstrated by conducting a concurrent multiscale simulation of a large-deformation manufacturing process with sub-grain spatial resolution while maintaining reasonable computational expense. This method could be used for microstructure-sensitive properties design as well as process parameters optimization.

During the lifetime of all livingmulticellular organisms, wounds in their tissues are frequently observed. The capability of closing those gaps is fundamental for a healthy development. If done deficiently, many diseases may occur from simple inflammation to tumor formation. The wound healing process in epithelial tissue occurs in three different stages. The first one is the assembly of a supra-cellular actomyosin cable and its migration towards the wound edge, triggered bybiochemicalprocesses in which calcium plays a distinctive role. How this process is orchestrated following damage remains unclear. Later, after its positioning, the cable contracts driving the tissue towards the gap and reducing the wound area. Finally, cell migration towards the interior of the wound ends up sealing the tissue. In this work, we make use of a mechanicalcontinuummodel for the first two stages in order to developed and 2Dfinite element simulationswithin a monolithically fully implicit implementation. The model for the actomyosin cable formation involves the coupling of transientcalcium ionstransport, with actin fibers and myosin motors recruitment and non-linear mechanical response of the tissue. The contraction stage, the active deformation of the previously formed actomyosin cable is taken into account. The relative motion of the myosin motors over the actin filaments is modeled so there exists an active tissue contraction in the direction of those fibers. Upon implementation, the model is capable of performing a wide range of biophysical situations reported experimentally, as we demonstrate in our numerical results. We have been able to rationalize through computational mechanics the firing of calcium in the wound right after damage infliction as well as the consequent formation of actin ring, reproducing nicely what has been reported in biological literature. Thereafter, the numerical model of acto-myosin contraction, fully integrated with the non-linear mechanics of the problem, correlates with the mechanics of wound closure at the actin-ring contraction stage. More importantly, the approach is the first of its kind in the modeling of epithelial and embryonic cell layers, where a wide number of complex mechanics has been integrated and solved though computational methods in engineering. We believe that the simulations will help to unravel new insights in open questions of developmental biology.

An inverse Mean-Field Homogenization (MFH) process is developed to improve the computational efficiency of non-linear stochastic multiscale analyzes by relying on a micro-mechanics model. First full-field simulations of composite Stochastic Volume Element (SVE) realizations are performed to characterize the homogenized stochastic behavior. The uncertainties observed in the non-linear homogenized response, which result from the uncertainties of their micro-structures, are then translated to an incremental-secant MFH formulation by defining the MFH input parameters as random effective properties. These effective input parameters, which correspond to the micro-structure geometrical information and to the material phases model parameters, are identified by conducting an inverse analysis from the full-field homogenized responses. Compared to the direct finite element analyzes on SVEs, the resulting stochastic MFH process reduces not only the computational cost, but also the order of uncertain parameters in the composite micro-structures, leading to a stochastic Mean-Field Reduced Order Model (MF-ROM). A data-driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic MF-ROM embedded in a Stochastic Finite Element Method (SFEM). The two cases of elastic Unidirectional (UD) fibers embedded in an elasto-plastic matrix and of elastic UD fibers embedded in a damage-enhanced elasto-plastic matrix are successively considered. In order to illustrate the capabilities of the method, the stochastic response of a ply is studied under transverse loading condition.

We present a stabilized finite element method that generalizes Nitsche’s method for enforcing stiff anisotropic cohesive laws with different normal and tangential stiffness. For smaller values of cohesive stiffness, the stabilized method resembles the standard method, wherein the traction on the crack surface is enforced as a Neumann boundary condition. Conversely, for larger values of cohesive stiffness, the stabilized method resembles Nitsche’s method, wherein the cohesive law is enforced as a kinematic constraint. We present several numerical examples, in two-dimensions, to compare the performance of the stabilized and standard methods. Our results illustrate that the stabilized method enables accurate recovery of crack-face tractions for stiff isotropic and anisotropic cohesive laws; whereas, the standard method is less accurate due to spurious traction oscillations. Also, the stabilized method could mitigate spurious sensitivity of load–displacement results to displacement increment in mixed-mode fracture simulation, owing to its stability and accuracy.

A novel FEM-Cluster based reduced order method or FEM-Cluster based Analysis method (FCA) which enables efficient numerical prediction of effective properties of heterogeneous material in nonlinear range is proposed. The cluster concept initially presented in the work by WK Liu et al. is introduced and extended to derive a full FEM multi-scale formulation of the Representative Unit Cell (RUC) to circumvent the heavy burden due to huge computational efforts required for a direct numerical simulation (DNS) of the high-fidelity RUC. The proposed FEM-Cluster based reduced order method is formulated in a consistent framework of finite element method . The offline clustering process with construction of the cluster-interaction matrix derived under the assumption of the linear elasticity is carried out by the devised FE procedure of RUC. The online elasto-plastic process is performed by the incremental non-linear FE analysis using the constant cluster-interaction matrix, which plays a role in the present work conceptually similar to the initial elastic modular matrix used in the ”initial stiffness method” for the traditional incremental elasto-plastic analysis. Accurate and efficient numerical prediction of effective properties of heterogeneous material in nonlinear range are developed in a consistent way. The performances of the proposed reduced order model and its numerical implementation are studied and demonstrated. Several numerical examples show its efficiency and applicability.

In this work a fully partitioned Lagrangian framework for the solution of fluid–structure interaction (FSI) problems involving free surfaces, large solid displacements and deformations, and strong added mass effects is presented. The fluid is simulated using the Particle Finite Element Method (PFEM), while Metafor, a large deformations nonlinear Finite Element code, is employed to simulate the solid part. The fully partitioned coupling is ensured through an Interface Quasi-Newton Inverse Least Squares (IQN-ILS) (Degroote et al., 2009) strategy to avoid added mass effects. The Lagrangian particle nature of the PFEM allows the simulation of problems involving free surfaces and very large solid displacements, usually difficult to achieve with traditional body-fitted CFD techniques. We show that owing to the generality of its formulation the PFEM can be used as is in the framework of fully partitioned FSI coupling schemes, where minimal information (i.e. loads and displacements at the FSI interface) is exchanged between the fluid and the solid solvers. More importantly, we demonstrate that a fully partitioned PFEM–FEM coupling based on the IQN-ILS strategy allows the simulation of a very large spectrum of FSI problems without incurring added-mass instabilities. The performance of the IQN-ILS coupling strategy in a fully Lagrangian framework is also assessed and compared to more traditional approaches such as Block-Gauss–Seidel (BGS) iterations with Aitken relaxation. An extensive work of verification and benchmarking is proposed, aiming to encompass all the combinations of physical and numerical parameters possibly leading to added-mass instabilities, and testing the IQN-ILS strategy on different benchmarks beyond those already proposed in the literature. The coupling is performed through CUPyDO (Thomas et al., 2019), a general Python framework for partitioned FSI coupling.

Numerical analysis of physical/mathematical problems based on generalized/extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices. Blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks. The presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed. In order to alleviate these drawbacks, three different Stable Generalized IsoGeometric Analysis (SGIGA) methods are proposed in this paper. In SGIGA, the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function. Problems with weak and strong discontinuities, singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed SGIGA methods. From the results, it is observed that SGIGA yields optimal convergence rates as well as better conditioning of system matrices. The results obtained from the proposed SGIGA methods are also compared with the results from the established methods, Stable Generalized Finite Element Method (SGFEM) and eXtended IsoGeometric Analysis (XIGA), to study the relative performances with respect to accuracy and conditioning.

We propose a method to obtain diagonal mass matrices for NURBS-based approximation spaces by a “dual lumping” method. The use of lumped mass matrices is of great importance in elastodynamics problems, as they can be employed in explicit time integration schemes which do not require the solution of a linear system. In finite elements, several well-established methods, such as row-sum, diagonal scaling, or nodal quadrature methods have been used to obtain lumped mass matrices for different applications. However, for higher-order and higher continuity approximation spaces such as those derived from NURBS, these approaches have only limited (second-order) accuracy. In this work, we derive a dual basis which has optimal approximation and dispersion properties, while maintaining local support. The dual space has discontinuities at the element boundaries (knots) and it is used to provide the test functions in the context of a Petrov–Galerkin method. This results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high-order accuracy. Numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity.