Fixed Background Mesh Research Articles

A Chimera method for thermal part-scale metal additive manufacturing simulation

This paper presents a Chimera approach for the thermal problems in welding and metallic Additive Manufacturing (AM). In particular, a moving mesh is attached to the moving heat source while a fixed background mesh covers the rest of the computational domain. The thermal field of the moving mesh is solved in the heat source reference frame. The chosen framework to couple the solutions on both meshes is a non-overlapping Domain Decomposition (DD) with Neumann–Dirichlet transmission conditions.Increased steadiness and accuracy within the vicinity of the Heat Affected Zone (HAZ) are the main advantages of this approach. The steadiness gain allows for the use of larger time steps, which is vital in AM applications and, in particular, Laser Powder Bed Fusion (LPBF), where the disparity of time scales represents a major hurdle. Moreover, enhanced accuracy can be observed in the resulting morphology of the melt pool. It will be shown that the method addresses classical shortcomings pointed out by Goldak without requiring the use of an asymmetrical heat source profile.

An embedded strategy for large scale incompressible flow simulations in moving domains

In this work we describe a methodology to approximate the incompressible Navier-Stokes equations in time dependent domains. To deal with the motion of the domain, we employ a fixed mesh method that we call fixed-mesh ALE. It consists of writing the equations in a moving ALE reference system but then projecting them onto a fixed background mesh. This implies that the boundaries of the elements do not necessarily coincide with the physical boundaries, and thus there is the possibility of badly cut elements. We use a Nitsche's type formulation to prescribe the boundary conditions and stabilise the bad cuts by introducing a term that penalises the gradient of the unknown orthogonal to the finite element space in a patch that contains the badly cut element. The flow formulation is a stabilised finite element method that allows one to treat convection dominated flows and to use equal velocity–pressure interpolation. Furthermore, this formulation can be shown to behave as an implicit large eddy simulation approach. A key issue is that the sub-grid scales on which the formulation depends are allowed to be time dependent; this fact has proved to be crucial for the robustness of the approach. The calculation of the velocity and the pressure is segregated by using a fractional step scheme designed at the pure algebraic level, and the conditioning of the pressure equation is improved by using an artificial compressibility technique. Finally, an adaptive mesh refinement strategy is described. All the algorithms are implemented in a parallel environment. The strategy described can be applied to complex flow problems, and in particular here we show the simulation of the air flow generated by a train moving inside a tunnel.

On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems

Problems with moving sources and moving inner boundaries in transient regime are of high interest in many research fields and engineering applications. One approach to properly tackle such problems is based on the Chimera method for non-matching grids, where each moving object is defined on a fine mesh that moves across the fixed coarse background. In this way, the high gradients around moving sources or boundaries are captured by the fine mesh without need of globally fine fixed meshes or adaptive refinement. In this work, Chimera is implemented in the framework of the finite element method, following an arbitrary Lagrangian–Eulerian formulation for the moving meshes and a standard Eulerian formulation for the fixed background mesh. Further, in case of convection-dominated problems, the scheme is stabilized using the streamline upwind Petrov–Galerkin method. The coupling between the moving fine meshes and the coarse fixed one is achieved via Dirichlet boundary conditions and a high-order interpolation algorithm. The performance of the proposed methodology in terms of accuracy and stability is assessed by means of numerical tests to be compared with equivalent problems solved using fixed meshes. Further tests serve to highlight the good performance of the proposed Chimera-based finite element method to address both convection-dominated problems with multiple moving boundaries and sources, and three-dimensional arc welding processes. • An algebraic-based Chimera-FEM scheme for unsteady problems is proposed. • Remeshing is avoided for large movements of heat sources and boundaries. • Advective-dominant problems can be addressed with Chimera-FEM and SUPG. • Good accuracy is obtained compared to conventional one-domain approaches. • The scheme is suitable to be used on 3D welding process simulations.

Embedded domain Reduced Basis Models for the shallow water hyperbolic equations with the Shifted Boundary Method

We consider fully discrete embedded finite element approximations for a shallow water hyperbolic problem and its reduced-order model. Our approach is based on a fixed background mesh and an embedded reduced basis. The Shifted Boundary Method for spatial discretization is combined with an explicit predictor/multi-corrector time integration to integrate in time the numerical solutions to the shallow water equations, both for the full and reduced-order model. In order to improve the approximation of the solution manifold also for geometries that are untested during the offline stage, the snapshots have been pre-processed by means of an interpolation procedure that precedes the reduced basis computation. The methodology is tested on geometrically parametrized shapes with varying size and position.

Random geometries for optimal control PDE problems based on fictitious domain FEMs and cut elements

This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually computationally “forbidden” combination of poorly conditioned equation system matrices due to challenging geometries, optimal control searches with iterative methods, slow convergence to system solutions on deterministic and non-deterministic level, and expensive remeshing due to geometrical changes. We overcome all these difficulties, utilizing the advantages of proper preconditioners adapted to unfitted mesh methods, improved types of Monte Carlo methods, and mainly employing the advantages of embedded FEMs, based on a fixed background mesh computed once even if geometrical changes are taking place. The sensitivity of the control problem is introduced in terms of random domains, employing a Quasi Monte Carlo method and emphasizing on a deterministic target state. The variational discretization concept is adopted, optimal error estimates for the state, adjoint state and control are derived that confirm the efficiency of the cut finite element method in challenging geometries. The performance of a multigrid scheme especially developed for unfitted finite element discretizations adapted to the optimal control problem is also tested. Some fundamental preconditioners are applied to the arising sparse linear systems coming from the discretization of the state and adjoint state variational forms in the spatial domain. The corresponding convergence rates along with the quality of the prescribed preconditioners are verified by numerical examples.

Discrete empirical interpolation and unfitted mesh FEMs: application in PDE-constrained optimization

In this work, we investigate the performance CutFEM as a high fidelity solver as well as we construct a competent and economical reduced order solver for PDE-constrained optimization problems in parametrized domains that live in a fixed background geometry and mesh. Its effectiveness and reliability will be assessed through its application for the numerical solution of quadratic optimization problems with elliptic equations as constraints, examining an archetypal case. The reduction strategy will be via Proper Orthogonal Decomposition of suitable FE snapshots, using an aggregated state and adjoint test space, while the efficiency of the offline-online decoupling will be ensured by means of Discrete Empirical Interpolation of the optimality system matrix and right-hand side, enabling thus a rapid resolution of the reduced order model for each new spatial configuration.

Simple, accurate, and efficient embedded finite element methods for fluid–solid interaction

This work presents a new approach to implementing a recently proposed optimal order Cut Finite Element Method (CutFEM) for problems with moving embedded solid structures in viscous incompressible flows. This new approach uses the notion of equivalent polynomials, introduced previously in the context of the eXtended Finite Element Methods (XFEM), to implement exact integration for terms involving products of polynomials with Heaviside and Dirac distributions. Combining CutFEM and equivalent polynomials results in a method for fluid–structure interaction that (1) has the same number of degrees of freedom as the underlying conforming Galerkin method on the fixed background mesh, which is independent of the configuration of non-conforming interfaces, (2) has the same element assembly structure as classical FEM on the background mesh—with standard quadrature rules, and (3) retains the convergence properties, indeed the precise theoretical structure, of the original CutFEM method. The result is a method that is robust, accurate, and efficient for interacting particles, and which provides a convenient approach for upgrading legacy finite element codes to include embedded boundaries with CutFEM or similar formulations that can retain the same asymptotic accuracy as the underlying boundary conforming finite element method.

Shape optimization of the time-harmonic response of vibroacoustic devices using cut elements

This article presents a method for generalized shape optimization of time-harmonic vibroacoustic problems. The modeling approach utilizes an immersed boundary cut element method in conjunction with a level set representation of the geometry. The cut element method utilizes a fixed background mesh, a dimensionless contrast parameter and an integration scheme to realize complex geometries and obtain accurate physical solutions to the governing problem. The design parameterization is obtained using a nodal level set description, directly linked to the mathematical design variables, and the gradients of the objective and the constraints are obtained with the discrete adjoint approach. The framework is applied to the optimization of three 2D examples. A study on the effect of initial guess for the proposed optimization procedure is presented on a benchmark example of the design of an acoustic partitioner. Further optimization examples include design of a wave splitter to realize prescribed frequency dependent directivity for emitted acoustic waves and a suspension structure design to improve the performance of a simplified 2D model of a hearing instrument. The results demonstrate that, even though the final topology is strongly dictated by the initial design, modifying the shape allows for a significant improvement of the system behavior. • A generalized shape optimization methodology for 2D time-harmonic vibroacoustic problems using cut elements. • A simple, yet accurate, crisp interface cut element method without the need for stabilization. • A numerically stable level set based shape optimization procedure based on density methods. • Three numerical examples that demonstrate the strengths and weaknesses of the proposed method.

A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems

We focus on steady and unsteady Navier–Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1–3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.

Numerical simulation of compressible fluid-particle flows in multimaterial Lagrangian hydrodynamics framework

In the numerical simulation of multimaterial flows, Lagrangian method was applied widely, and fluid-particle multiphase flows phenomenon often existed when elastic-plastic material was impacted intensively. Most of the previous studies about the fluid-particle flows in Lagrangian hydrodynamics framework only focused on the dilute flow. It is very necessary to establish a Lagrangian-Lagrangian coupling numerical simulation framework, which is suitable for both multimaterial and dense particle flows in engineering applications. In present study, a numerical model, in which both continuous fluid phase and dispersed particle phase are discretized in Lagrangian frameworks, is developed. In this model, the effect of particle volume fraction, collision force among particles, interaction and coupling between fluid and particle phases are all considered based on the physical meanings. Therefore it is well-suited to complex multiphase flows including transition between dense and dilute particles flows, and material interfaces and free boundaries in multimaterial flows can be solved spontaneously. The relevant numerical technology and approach are proposed and derived, including fixed background mesh, higher order interpolation function, discetezation of coupling source terms and so on. The above numerical model and approach are integrated to a multimaterial staggered-grid elastic-plastic Lagrangian hydrodynamics code. A series of tests, including single particle transport, shock-induced dense particles bed, and explosive dispersal of solid particles, are used to validate the model. The numerical results show excellent agreement with theoretical solutions or experimental measurements, providing confidence in the proposed numerical model and methods.

Generalized shape optimization of transient vibroacoustic problems using cut elements

AbstractThis article propose a generalized shape optimization method for transient coupled acoustic–mechanical interaction problems. The transient problem formulation allows to optimize for a broadband frequency content and the possibility to investigate transient phenomena such as pulse shaping. Throughout the work, the geometry is defined with the help of a nodal‐based design field, for which its zero‐level contour describes the interface between acoustic and structural domains. The approach utilizes a fixed background mesh to represent the geometry and a cut element immersed boundary method for the physical modeling. This modeling approach provides accurate solutions to the strongly coupled governing equations based on a special integration scheme. The optimization problem is solved using a gradient‐based optimizer and employs a fully discrete adjoint approach for calculating the sensitivity information. A numerical examination on the accuracy of the sensitivity analysis compares the fully discrete gradients to the commonly used semidiscrete adjoint approach for transient optimization revealing the importance of consistent sensitivities. The transient design formulation is validated on a 2D benchmark problem concerning the design of a time‐harmonic acoustic partitioner. The developed framework is then applied for the design of vibroacoustic pulse shaping devices to demonstrate control of a transient phenomena.

A manifold Lagrangian integration point method for large deformation failure modelling of rock slopes

A manifold Lagrangian integration point method is proposed by combining the manifold coverage algorithm and a Lagrangian integral point finite element method, aiming to simulate the large deformation failure in slope engineering. The model is established based on the dual-discrete method with a fixed Eulerian background mesh and moving material particles. The method adopts the Eulerian mesh as the fixed background mesh and describes the deformation behaviour of macroscopic objects via particle motion between meshes, thus avoiding the problem of grid distortion in the calculation process and eliminating particle motion constraints in the set region. A strain softening strain energy density failure criterion is introduced for the first time to determine the failure of the continuum, such that the process of discontinuous failure can be addressed by a continuous method. Finally, the method is verified by simulating the post-excavation failure evolution of a slope engineering case for highway reconstruction and expansion. The results found that the obtained slope slip surface is essentially consistent with the theoretical solution.

Optimization of an internal blade cooling passage configuration using a Chimera approach and parallel computing

Improving gas turbine performance depends almost exclusively on the maximum temperature of the combustion gases. This temperature is limited by the thermomechanical strength of the turbine vanes. In this work, a Chimera approach for overlapping grids in the finite element method (FEM) context is proposed to optimize the arrangement of several cooling passages within the vane to minimize its average temperature, thereby improving the thermal transfer efficiency. The scheme is based on a fixed background mesh covering the entire domain, while finer meshes surrounding the coolant passages can move around over the airfoil while searching for the best configuration of the cooling system. Information is exchanged between meshes through a high-order interpolation algorithm. The optimization strategy is developed based on the use of the population-based augmented Lagrangian particle swarm optimizer (ALPSO) to approach a probable global minimum, and then a refinement of the solution is performed using the gradient-based sequential least squares quadratic programming (SLSQP) algorithm. Both the gradient, for the gradient-based optimizer, and the function evaluations, for the population-based optimizer, are solved in parallel to speed up the solution.

An extended/generalized phase‐field finite element method for crack growth with global‐local enrichment

SummaryAn extended/generalized finite element method (XFEM/GFEM) for simulating quasistatic crack growth based on a phase‐field method is presented. The method relies on approximations to solutions associated with two different scales: a global scale, that is, structural and discretized with a coarse mesh, and a local scale encapsulating the fractured region, that is, discretized with a fine mesh. A stable XFEM/GFEM is employed to embed the displacement and damage fields at the global scale. The proposed method accommodates approximation spaces that evolve between load steps, while preserving a fixed background mesh for the structural problem. In addition, a prediction‐correction algorithm is employed to facilitate the dynamic evolution of the confined crack regions within a load step. Several numerical examples of benchmark problems in two‐ and three‐dimensional quasistatic fracture are provided to demonstrate the approach.

Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions

We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represent parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche’s method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.

An immersed discontinuous finite element method for the Stokes problem with a moving interface

Abstract We present a discontinuous immersed finite element (IFE) method for incompressible interfacial flows that are governed by the Stokes equations. The method is based on a Cartesian mesh with elements cut by the moving interface. On this fixed unfitted mesh, we employ an immersed Q 1 ∕ Q 0 finite element space constructed according to the location of the interface and pertinent interface jump conditions. As such, the smearing of solution across the interface is greatly reduced. The interface, represented by a sequence of marker points, is advected on the fixed background mesh by the local fluid velocity. The mesh is locally refined near the interface to further improve accuracy. Compared with the phase-field method on adaptive meshes, our method can achieve the same level of accuracy with much less degrees of freedoms. We present some numerical examples to validate and demonstrate the capability of the proposed method.

Cut finite elements for convection in fractured domains

We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain, which is a union of manifolds of different dimensions such that a d dimensional component always resides on the boundary of a d+1 dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

Universal meshes for a branched crack

We introduce an efficient approach to obtain conforming meshes for evolving branched cracks immersed in a fixed background mesh. The proposed approach is built on universal meshes (UM) proposed in Rangarajan et al. [34] which is able to construct conforming triangulations for a propagating simple crack. The UM functions by projecting certain nodes of the background mesh onto the crack and simultaneously relaxing their neighboring nodes to improve the quality of the resulting triangular mesh. The essence of the generalization to a branched crack is to determine which side of each branch to select nodes to move to the crack path. The choice is based on the consideration of minimizing mesh distortion. For the case of multiple junctions, we take into account the constraint that the nodes to be moved to the same crack branch must be on the same side of that branch. The proposed method inherits the main advantages of UM, including small perturbation to a fixed background mesh for a family of evolving cracks with no a priori conformity requirements. This advantage saves computational time compared with a brute-force mesh generation step. Numerical examples with one or multiple triple or quadruple junctions are provided.

A cut finite element method for the Bernoulli free boundary value problem

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H1 norm. Finally, we present illustrating numerical results.

A new cut-cell algorithm for DSMC simulations of rarefied gas flows around immersed moving objects

Direct Simulation Monte Carlo (DSMC) is a widely applied numerical technique to simulate rarefied gas flows. For flows around immersed moving objects, the use of body fitted meshes is inefficient, whereas published methods using cut-cells in a fixed background mesh have important limitations. We present a novel cut-cell algorithm, which allows for accurate DSMC simulations around arbitrarily shaped moving objects. The molecule–surface interaction occurs exactly at the instantaneous collision point on the moving body surface, and accounts for its instantaneous velocity, thus precisely imposing the desired boundary conditions. A simple algorithm to calculate the effective volume of cut cells is presented and shown to converge linearly with grid refinement. The potential and efficiency of method is demonstrated by calculating rarefied gas flow drag forces on steady and moving immersed spheres. The obtained results are in excellent agreement with results obtained with a body-fitted mesh, and with analytical approximations for high-Knudsen number flows.