Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics

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

Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization, outlier removal and robust time integration

In this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods based on \cite{gebhardt_2021_beam} that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space $\left(\mathbb{R}^3\right)$, which is larger than the corresponding multiple copies of the manifold \left(\mathbb{R}^3 \cross S^2\right) obtained with standard Hermite finite elements. For implicit time integration, we choose a hybrid form of the mid-point rule and the trapezoidal rule that preserves the linear angular momentum exactly and approximates the energy accurately. In addition, we apply a recently introduced approach for outlier removal \cite{hiemstra_outlier_2021} that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines

Consensus molecular subtype classification of colorectal adenomas

Consensus molecular subtyping is an RNA expression-based classification system for colorectal cancer (CRC). Genomic alterations accumulate during CRC pathogenesis, including the premalignant adenoma stage, leading to changes in RNA expression. Only a minority of adenomas progress to malignancies, a transition that is associated with specific DNA copy number aberrations or microsatellite instability (MSI). We aimed to investigate whether colorectal adenomas can already be stratified into consensus molecular subtype (CMS) classes, and whether specific CMS classes are related to the presence of specific DNA copy number aberrations associated with progression to malignancy. RNA sequencing was performed on 62 adenomas and 59 CRCs. MSI status was determined with polymerase chain reaction-based methodology. DNA copy number was assessed by low-coverage DNA sequencing (n = 30) or array-comparative genomic hybridisation (n = 32). Adenomas were classified into CMS classes together with CRCs from the study cohort and from The Cancer Genome Atlas (n = 556), by use of the established CMS classifier. As a result, 54 of 62 (87%) adenomas were classified according to the CMS. The CMS3 ‘metabolic subtype’, which was least common among CRCs, was most prevalent among adenomas (n = 45; 73%). One of the two adenomas showing MSI was classified as CMS1 (2%), the ‘MSI immune’ subtype. Eight adenomas (13%) were classified as the ‘canonical’ CMS2. No adenomas were classified as the ‘mesenchymal’ CMS4, consistent with the fact that adenomas lack invasion-associated stroma. The distribution of the CMS classes among adenomas was confirmed in an independent series. CMS3 was enriched with adenomas at low risk of progressing to CRC, whereas relatively more high-risk adenomas were observed in CMS2. We conclude that adenomas can be stratified into the CMS classes. Considering that CMS1 and CMS2 expression signatures may mark adenomas at increased risk of progression, the distribution of the CMS classes among adenomas is consistent with the proportion of adenomas expected to progress to CRC

A matrix‐free macro‐element variant of the hybridized discontinuous Galerkin method

We investigate a macro‐element variant of the hybridized discontinuous Galerkin (HDG) method, using patches of standard simplicial elements that can have non‐matching interfaces. Coupled via the HDG technique, our method enables local refinement by uniform simplicial subdivision of each macro‐element. By enforcing one spatial discretization for all macro‐elements, we arrive at local problems per macro‐element that are embarrassingly parallel, yet well balanced. Therefore, our macro‐element variant scales efficiently to n‐node clusters and can be tailored to available hardware by adjusting the local problem size to the capacity of a single node, while still using moderate polynomial orders such as quadratics or cubics. Increasing the local problem size means simultaneously decreasing, in relative terms, the global problem size, hence effectively limiting the proliferation of degrees of freedom. The global problem is solved via a matrix‐free iterative technique that also heavily relies on macro‐element local operations. We investigate and discuss the advantages and limitations of the macro‐element HDG method via an advection‐diffusion model problem

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the consistent mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. We perform numerical experiments for the linear wave equation in one and two dimensions, for quadrilateral elements and triangular elements, and for up to fourth order polynomial basis functions. Despite the increase in critical time-step size, we do not observe adverse effects in terms of spatial accuracy and orders of convergence. To extend the method to non-linear problems, we introduce a linear approximation. Our three-dimensional experiments with tetrahedral and hexahedral elements show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.</p

A variational approach based on perturbed eigenvalue analysis for improving spectral properties of isogeometric multipatch discretizations

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, optical branches of spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces. In this paper, we introduce a variational approach based on perturbed eigenvalue analysis that eliminates outlier frequencies without negatively affecting the accuracy in the remainder of the spectrum and modes. We then propose a pragmatic iterative procedure that estimates the perturbation parameters in such a way that the outlier frequencies are effectively reduced. We demonstrate that our approach allows for a much larger critical time-step size in explicit dynamics calculations. In addition, we show that the critical time-step size obtained with the proposed approach does not depend on the polynomial degree of spline basis functions.</p

Modeling of Growth using an Immersed Finite Element Method

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