Computation in Engineering

In the last decade a family of methods called meshless methods has been developed both for structural and fluid mechanics problems. After these ideas, a possible classification for numerical formulations may be to separate the methods that make use of a standard finite element mesh (such as those made of tetrahedra or hexahedra), from those that do not need a standard mesh, namely the meshless methods. For solving a partial different equation by a numerical method, a possible alternative may be either to use a mesh method or a meshless method. This paper discusses this issue to show that this choice is not, in the large majorities of the cases, the right question.

This paper presents an application of the subset simulation method to the finite element reliability assessment of a deep circular tunnel. The tunnel is assumed to be surrounded by weak rock with uncertain material properties. For the spatial variability of the uncertain parameters the midpoint method is utilized. Depending on the assumed correlation length, the above stochastic discretization method may lead to a very large number of random variables. In such cases, most Monte Carlo variants prove to be inefficient in computing small failure probabilities, which are typical in structural engineering. However, the subset simulation method overcomes this problem by expressing the failure probability as a product of larger conditional probabilities, which are estimated by applying an efficient Markov chain Monte Carlo technique. The reliability analysis is performed using a reliability tool, which is being developed as part of the SOFiSTiK finite element software package.

In this contribution, the finite cell method (FCM) is applied to solve transient problems of linear elastodynamics. The mathematical formulation of FCM for linear elastodynamics is presented, following from the weak formulation of the initial/boundary-value problem. Semidiscrete time integration schemes are briefly discussed, and the choice of implicit time integration is justified. A one-dimensional benchmark problem is solved using FCM, illustrating the method's ability to solve problems of linear elastodynamics obtaining high rates of convergence. Furthermore, a numerical example of transient analysis from an industrial application is solved using FCM. The numerical results are compared to the results obtained using state-of-the-art commercial software, employing linear finite elements, in conjunction with explicit time integration. The results illustrate the potential of FCM as a powerful tool for transient analysis in elastodynamics, offering a high degree of accuracy at a moderate computational effort.

The implementation of hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most hp-code frameworks restrict themselves to 1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the hp-d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence-both in terms of degrees of freedom and in run-time-for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic hp-adaptivity.

Contact problems in solid mechanics are traditionally solved using the h-version of the finite element method. The constraints are enforced along the surfaces of e.g. elastic bodies under consideration. Standard constraint algorithms include penalty methods, Lagrange multiplier methods and combinations thereof. For complex scenarios, a major part of the solution time is taken up by operations to identify points that come into contact. This paper presents a novel approach to model frictionless contact using high order finite elements. Here, we employ an especially designed material model that is inserted into two-respectively threedimensional regions surrounding contacting bodies. Contact constraints are thus enforced on the same manifold as the accompanying structural problem. The application of the current material formulation leads to a regularization of the Karush-Kuhn-Tucker conditions. Our formulation can be classified as a barrier-type method. Results are obtained for two-and three-dimensional problems, including a Hertzian contact problem. Comparisons to a commercial FEA package are provided. The proposed formulation works well for non-matching discretizations on adjacent contact interfaces and handles self-contact naturally. Since the non-penetrating conditions are solved in a physically consistent manner, there is no need for an explicit contact search.

One of the essential steps of performing a finite element analysis is the discretisation of the geometric domain into an analysis-suitable mesh. In case of complex geometries this process may become a severe bottleneck in the complete analysis pipeline. Studies show that the time required for creating an analysis-suitable geometry and its computational mesh accounts for about 80% of the overall analysis time [1].

One of the essential steps of performing a finite element analysis is the discretisation of the geometric domain into an analysis-suitable mesh. In case of complex geometries this process may become a severe bottleneck in the complete analysis pipeline. Studies show that the time required for creating an analysis-suitable geometry and its computational mesh accounts for about 80% of the overall analysis time [1].

During the last ten years, increasing efforts were made to improve and simplify the process from Computer Aided Design (CAD) modeling to a numerical simulation. It has been shown that the transition from one model to another, i.e. the meshing, is a bottleneck. Several approaches have been developed to overcome this time-consuming step, e.g. Isogeometric Analysis (IGA), which applies the shape functions used for the geometry description (typically B-Splines and NURBS) directly to the numerical analysis. In contrast to IGA, which deals with boundary represented models (B-Rep), our approach focuses on parametric volumetric models such as Constructive Solid Geometries (CSG). These models have several advantages, as their geometry description is inherently watertight and they provide a description of the models interior. To be able to use the explicit mathematical description of these models, we employ the Finite Cell Method (FCM). Herein, the only necessary input is a reliable statement whether an (integration-) point lies inside or outside of the geometric model. This paper mainly discusses such point-in-membership tests on various geometric objects like sweeps and lofts, as well as several geometric operations such as filleting or chamfering. We demonstrate that, based on the information of the construction method of these objects, the point-in-membership-test can be carried out efficiently and robustly.

Articular cartilage from a material science point of view is a soft network composite that plays a critical role in load-bearing joints during dynamic loading. Its composite structure, consisting of a collagen fiber network and a hydrated proteoglycan matrix, gives rise to the complex mechanical properties of the tissue including viscoelasticity and stress relaxation. Melt electrospinning writing allows the design and fabrication of medical grade polycaprolactone (mPCL) fibrous networks for the reinforcement of soft hydrogel matrices for cartilage tissue engineering. However, these fiber-reinforced constructs underperformed under dynamic and prolonged loading conditions, suggesting that more targeted design approaches and material selection are required to fully exploit the potential of fibers as reinforcing agents for cartilage tissue engineering. In the present study, we emulated the proteoglycan matrix of articular cartilage by using highly negatively charged star-shaped poly(eth...

Recently, a multi-level hp-version of the finite element method (FEM) was proposed to ease the difficulties of treating hanging nodes, while providing full hp-approximation capabilities. In the original paper, the refinement procedure made use of a-priori knowledge of the solution. However, adaptive procedures can produce discretizations which are more effective than an intuitive choice of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable. The present contribution demonstrates that multi-level hp-adaptive schemes can be efficiently driven by an explicit a-posteriori error estimator. To this end, we adopt the classical residual-based error estimator. The main insight here is that its extension to multi-level hp-FEM is possible by considering the refined-most overlay elements as integration domains. We demonstrate on several two-and three-dimensional examples that exponential convergence rates can be obtained.

The multi-level hp-refinement scheme is a powerful extension of the finite element method that allows local mesh adaptation without the trouble of constraining hanging nodes. This is achieved through hierarchical high-order overlay meshes, a hp-scheme based on spatial refinement by superposition. An efficient parallelization of this method using standard domain decomposition approaches in combination with ghost elements faces the challenge of a large basis function support resulting from the overlay structure and is in many cases not feasible. In this contribution, a parallelization strategy for the multi-level hp-scheme is presented that is adapted to the scheme's simple hierarchical structure. By distributing the computational domain among processes on the granularity of the active leaf elements and utilizing shared mesh data structures, good parallel performance is achieved, as redundant computations on ghost elements are avoided. We show the scheme's parallel scalability for problems with a few hundred elements per process. Furthermore, the scheme is used in conjunction with the finite cell method to perform numerical simulations on domains of complex shape.

This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with "dirty geometries". FCM, being a fictitious domain approach based on higher order finite elements, embeds the physical model into a fictitious domain, which can be discretized without having to take into account the boundary of the physical domain. The true geometry is captured by a precise numerical integration of elements cut by the boundary. Thus, an effective Point Membership Classification algorithm that determines the inside-outside state of an integration point with respect to the physical domain is a core operation in FCM. To treat also "dirty geometries", i.e. imprecise or flawed geometric models, a combination of a segment-triangle intersection algorithm and a flood fill algorithm being insensitive to most CAD model flaws is proposed to identify the affiliation of the integration points. The present method thus allows direct computations on geometrically and topologically flawed models. The potential and merit for practical applications of the proposed method is demonstrated by several numerical examples.

In this contribution, we validate a physical model based on a transient temperature equation (including latent heat) w.r.t. the experimental set AMB2018-02 provided within the additive manufacturing benchmark series, established at the National Institute of Standards and Technology, USA. We aim at predicting the following quantities of interest: width, depth, and length of the melt pool by numerical simulation and report also on the obtainable numerical results of the cooling rate. We first assume the laser to posses a double ellipsoidal shape and demonstrate that a well calibrated, purely thermal model based on isotropic thermal conductivity is able to predict all the quantities of interest, up to a deviation of maximum 7.3% from the experimentally measured values. However, it is interesting to observe that if we directly introduce, whenever available, the measured laser profile in the model (instead of the double ellipsoidal shape) the investigated model returns a deviation of 19.3% from the experimental values. This motivates a model update by introducing anisotropic conductivity, which is intended to be a simplistic model for heat material convection inside the melt pool. Such an anisotropic model enables the prediction of all quantities of interest mentioned above with a maximum deviation from the experimental values of 6.5%. We note that, although more predictive, the anisotropic model induces only a marginal increase in computational complexity.

In this paper, the use of ℎ-basis functions with higher differentiability properties is discussed in the context of the finite cell method and numerical simulations on complex geometries. For this purpose, ℎ-basis functions based on classical B-splines and a new approach for the construction of 1 ℎ-basis functions with minimal local support are introduced. Both approaches allow for hanging nodes, whereas the new 1 approach also includes varying polynomial degrees. The properties of the ℎ-basis functions are studied in several numerical experiments, in which a linear elastic problem with some singularities is discretized with adaptive refinements. Furthermore, the application of the ℎ-basis functions based on B-splines is investigated in the context of nonlinear material models, namely hyperelasticity and elastoplasicity with finite strains. K E Y W O R D S continuity, finite cell method, finite strain elastoplasticity, hierarchical B-splines, ℎ-basis, local refinement 1 INTRODUCTION The progress of industrial production techniques needs to be supported by suitable numerical simulation schemes. In this respect, additive manufacturing technologies enable an unprecedented design freedom, which is now less restricted by the manufacturing process and more driven by functionality. This results in artifacts of increasingly complex shapes, for example, presenting internal details, curved elements, enclosed cavities of irregular shape, or a porous-like structure. The ability to produce suitable computational analysis models for these kinds of structures is one of today's challenges in numerical simulation. The finite cell method (FCM) is designed to capture the mechanical behavior of a body using a mesh which is unfitted to its physical boundary with high accuracy and, thus, enables simulations on complex shapes with challenging topologies. The FCM has been initially proposed [1-3] as an immersed boundary method. Its high accuracy is obtained by combining the fictitious domain approach with high-order basis functions. The method has been applied successfully to a variety of problems in solid mechanics such as thermo-elasticity, [4] geometrical nonlinearities, [5] explicit and implicit elastodynamics, [6, 7] bio-mechanics, [8, 9] elastoplasticity, [10, 11] and foamed materials. [12, 13] Basis functions with higher differentiability properties are naturally provided in isogeometric analysis (IGA), since in this approach the description of the geometry and the discretization of the unknown field are implemented with the same basis This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.