Distributed Electric Propulsion systems are an emerging technology. Aerodynamic interactions between propellers in close proximity can, however, cause periodic variations in the blade loading. Together with acoustic interference, these installation effects can form a dominant noise source in such systems. In this contribution, we investigate a low-cost computational modeling approach to predict the unsteady loading of the propeller blades, and thereby the interaction noise of an array of side-by-side propellers. To inform this low-cost model, a numerical campaign of scale-resolving Lattice Boltzmann/Very Large Eddy Simulations (LBM/VLES) has been performed on the Dutch National Supercomputer Snellius. The goal of this model development is to gain a better understanding of the blade-to-blade interaction mechanisms and to determine to which extent the model can be applied for purposes like preliminary design, uncertainty quantification, or control, for which the computational cost of high-fidelity simulations is prohibitive. As a practical example, the optimal relative phase angle in an array of propellers is determined and validated.@en

Distributed electric propulsion systems are an emerging technology with the potential of revolutionizing the design and performance of aircraft. When propellers are located in close proximity, they can be subjected to aerodynamic interactions, which affect the far-field noise. In this paper, we study an array of three co-rotating and adjacent propellers to describe both the aerodynamic and acoustic installation effects. A scale-resolving CFD simulation based on the Lattice-Boltzmann/Very-Large-Eddy-Simulation method is used to solve the flow field around the propellers. An acoustic analogy integral approach calculates the far-field noise. Findings show that the helical vortical structures, generated at the tip of each blade undergo a flow deformation at the location of interaction. This causes the loading of each blade to vary during the rotation. Consequently, the unsteady loading noise becomes a dominant noise generation mechanism, driving the noise levels and directivity. It is also shown that introducing a non-zero relative phase angle between the propellers results in a reduction of the unsteady thrust, leading to a mitigation of the unsteady-loading tonal components along the rotation axis. Additionally, the relative phase angle causes constructive/destructive acoustic interference, as demonstrated by analyzing the noise emitted simultaneously by the three propellers.@en

This review paper discusses the developments in immersed or unfitted finite element methods over the past decade. The main focus is the analysis and the treatment of the adverse effects of small cut elements. We distinguish between adverse effects regarding the stability and adverse effects regarding the conditioning of the system, and we present an overview of the developed remedies. In particular, we provide a detailed explanation of Schwarz preconditioning, element aggregation, and the ghost penalty formulation. Furthermore, we outline the methodologies developed for quadrature and weak enforcement of Dirichlet conditions, and we discuss open questions and future research directions.@en

This chapter reviews the work conducted by our team on scan-based immersed isogeometric analysis for flow problems. To leverage the advantageous properties of isogeometric analysis on complex scan-based domains, various innovations have been made: (i) A spline-based segmentation strategy has been developed to extract a geometry suitable for immersed analysis directly from scan data; (ii) A stabilized equal-order velocity-pressure formulation for the Stokes problem has been proposed to attain stable results on immersed domains; (iii) An adaptive integration quadrature procedure has been developed to improve computational efficiency; (iv) A mesh refinement strategy has been developed to capture small features at a priori unknown locations, without drastically increasing the computational cost of the scan-based analysis workflow. We review the key ideas behind each of these innovations, and illustrate these using a selection of simulation results from our work. A patient-specific scan-based analysis case is reproduced to illustrate how these innovations enable the simulation of flow problems on complex scan data.

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In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric explicit dynamics context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche's method. The conventional stability requirement for Nitsche's method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of ‘mass-scaling’ does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff–Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

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We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation and adaptivity procedure are applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated into a scan-based analysis workflow, capable of generating error-controlled results from scan data without the need for extensive user interactions or interventions.

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This contribution presents a hierarchical multigrid approach for the solution of large-scale finite cell problems on both uniform grids and multi-level hp-discretizations. The proposed scheme takes advantage of the hierarchical basis functions utilized in the finite cell method and the multi-level hp-method, which is attributed to the use of high-order integrated Legendre basis functions and overlay meshes, to yield a simple and elegant multigrid scheme. This simplicity is reflected in the fact that transitioning between multigrid levels only involves the inclusion or exclusion of specific basis functions. All restriction and prolongation operators, therefore, reduce to binary matrices that do not need to be explicitly assembled or applied, saving computational time and memory. Elementwise and patchwise additive Schwarz smoothing techniques are used to mitigate the influence of the cut cells on the conditioning of the linear systems, while maintaining the parallelizability of the solver. The effectiveness of the scheme is numerically verified in various examples and convergence rates that are independent of the cut configuration, mesh size, refinement level and, in certain scenarios, even the polynomial order are shown. A series of numerical examples demonstrate the applicability of the scheme for solving large immersed systems with multiple millions and even billions of unknowns on massively parallel machines.

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We show that in the variational multiscale framework, the weak enforcement of essential boundary conditions via Nitsche's method corresponds directly to a particular choice of projection operator. The consistency, symmetry and penalty terms of Nitsche's method all originate from the fine-scale closure dictated by the corresponding scale decomposition. As a result of this formalism, we are able to determine the exact fine-scale contributions in Nitsche-type formulations. In the context of the advection–diffusion equation, we develop a residual-based model that incorporates the non-vanishing fine scales at the Dirichlet boundaries. This results in an additional boundary term with a new model parameter. We then propose a parameter estimation strategy for all parameters involved that is also consistent for higher-order basis functions. We illustrate with numerical experiments that our new augmented model mitigates the overly diffusive behavior that the classical residual-based fine-scale model exhibits in boundary layers at boundaries with weakly enforced essential conditions.

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Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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The finite cell method is a flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a wide range of geometrical models can be performed. The application of the finite cell method, and other immersed methods, to large real-life and industrial problems is often limited due to the conditioning problems associated with these methods. These conditioning problems have caused researchers to resort to direct solution methods. This significantly limits the maximum size of solvable systems. Iterative solvers are better suited for large-scale computations than their direct counterparts due to their lower memory requirements and suitability for parallel computing. These benefits can, however, only be exploited when systems are properly conditioned. In this contribution we present an Additive-Schwarz type preconditioner that enables efficient and parallel scalable iterative solutions of large-scale multi-level hp-refined finite cell systems.

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Immersed finite element methods generally suffer from conditioning problems when cut elements intersect the physical domain only on a small fraction of their volume. We present a dedicated Additive-Schwarz preconditioner that targets the underlying mechanism causing the ill-conditioning of these methods. This preconditioner is applicable to problems that are not symmetric positive definite and to mixed problems. We provide a motivation for the construction of the Additive-Schwarz preconditioner, and present a detailed numerical investigation into the effectiveness of the preconditioner for a range of mesh sizes, isogeometric discretization orders, and partial differential equations, among which the Navier–Stokes equations.

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Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. Of the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors.

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The (Isogeometric) Finite Cell Method–in which a domain is immersed in a structured background mesh–suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two- and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method.

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