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We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is ``close'' to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

To prevent remeshing, we explore the use of a non‐boundary‐fitted finite element method for the computational modeling of growth including contact mechanics. Accordingly, we utilize a mesh‐related mapping procedure for the use of implicit geometry description by a level set function within the framework of immersed methods. Hence, our framework provides a setting to include patient‐specific geometries based on imaging data as we use a level set function for the implicit geometry description. In this contribution, we show that the proposed approach is a viable alternative for problems with mesh‐related obstacles, in particular when large growth simulations on complex patient‐specific geometries are of primary interest.

Numerical computation of the Karhunen--Lo\`eve expansion is computationally challenging in terms of both memory requirements and computing time. We compare two state-of-the-art methods that claim to efficiently solve for the K--L expansion: (1) the matrix-free isogeometric Galerkin method using interpolation based quadrature proposed by the authors in [1] and (2) our new matrix-free implementation of the isogeometric collocation method proposed in [2]. Two three-dimensional benchmark problems indicate that the Galerkin method performs significantly better for smooth covariance kernels, while the collocation method performs slightly better for rough covariance kernels.

The Karhunen-Lo\`eve series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise $L^2$-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a $2d$ dimensional domain, where $d$, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix-vector product for iterative eigenvalue solvers. Two higher-order three-dimensional benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

Quadrilateral layouts on surfaces are valuable in texture mapping, and essential in generation of quadrilateral meshes and in fitting splines. Previous work has characterized such layouts as a special metric on a surface or as a meromorphic quartic differential with finite trajectories. In this work, a surface quadrilateral layout is alternatively characterized as a special immersion of a cut representation of the surface into the Euclidean plane. We call this a quad layout immersion. This characterization, while posed in smooth topology, naturally generalizes to piecewise-linear representations. As such, it mathematically describes and generalizes integer grid maps, which are common in computer graphics settings. Finally, the utility of the representation is demonstrated by computationally extracting quadrilateral layouts on surfaces of interest., Comment: 48 pages (31 for article, 17 for supplementary background material and appendices), 25 figures

Efficient product development constitutes a workflow in which design, analysis, optimization, and manufacturing all work in unison to develop, test, measure, refine and validate a product from first concept through detailed design to final prototype. There are several key challenges, however, that have hampered such a workflow for decades. The main bottleneck is the lack of a single, conforming representation of geometry at all stages of the design process enabling true interoperability across computer aided design (CAD), analysis (CAE) and manufacturing (CAM). Isogeometric analysis was introduced to improve the interoperability between computer aided design and finite element analysis by unifying the underlying technology that supports both disciplines. Its promise is to eliminate the tedious process of geometry repair, feature removal, and mesh generation, while, at the same time, maintaining a single exact representation of geometry at all stages of the design process. Although isogeometric analysis has proven its fidelity as an analysis technology, some key issues remain. Specifically, * Current CAD technologies describe geometry by means of the boundary representation or simply B-rep. Boolean operations, ubiquitous in computer aided design, use a process called trimming that leads to a non-conforming description of geometry that is un-editable and incompatible with all downstream applications, thereby inhibiting true interoperability across the design-through-analysis process. * The high cost of formation of element arrays and assembly into the global system limits isogeometric analysis to low polynomial orders, typically degree two or three. The same may be said for classical finite element analysis. Classical formation and assembly routines scale as O(p⁹) per degree of freedom, while the optimal scaling with polynomial degree is O(p³). The main objective of this work is the development of several supporting technologies that address these two issues and thereby enable stable, accurate and efficient high polynomial-order isogeometric analysis for applications in solid mechanics.

We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.