Simultaneous Shape and Mesh Quality Optimization using Pre-Shape Calculus
Authors: Daniel Luft, Volker Schulz
Abstract: Computational meshes arising from shape optimization routines commonly suffer from decrease of mesh quality or even destruction of the mesh. In this work, we provide an approach to regularize general shape optimization problems to increase both shape and volume mesh quality. For this, we employ pre-shape calculus (cf. arXiv:2012.09124). Existence of regularized solutions is guaranteed. Further, consistency of modified pre-shape gradient systems is established. We present pre-shape gradient system modifications, which permit simultaneous shape optimization with mesh quality improvement. Optimal shapes to the original problem are left invariant under regularization. The read morecomputational burden of our approach is limited, since additional solution of possibly larger (non-)linear systems for regularized shape gradients is not necessary. We implement and compare pre-shape gradient regularization approaches for a hard to solve 2D problem. As our approach does not depend on the choice of metrics representing shape gradients, we employ and compare several different metrics.
Pre-Shape Calculus: Foundations and Application to Mesh Quality Optimization
Authors: Daniel Luft, Volker Schulz
Abstract: Deformations of the computational mesh arising from optimization routines usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize classical shape optimization and calculus. We define pre-shape derivatives and derive according structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives featuring only normal directions. Techniques from classical shape optimization and -calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which read moreis solvable by use of pre-shape derivatives. This class allows for simultaneous optimization of classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.
Discrete port-Hamiltonian Coupled Heat Transfer
Authors: Jens Jäschke, Matthias Ehrhardt, Michael Günther, Birgit Jacob
Abstract: Heat transfer and cooling solutions play an important role in the design of gas turbine blades. However, the underlying mathematical coupling structures have not been thoroughly investigated. In a previous work, we successfully modeled a simplified version of this problem as an infinite-dimensional system. Here, we construct a spatial discretization for the above problem and investigate its properties. We show that the discrete system is less restrictive than the original infinite-dimensional system, suggesting something like a regularization effect due to discretization.
A Port-Hamiltonian Formulation of Coupled Heat Transfer
Authors: Jens Jäschke, Matthias Ehrhardt, Michael Günther, Birgit Jacob
Abstract: Heat transfer and cooling solutions play an important role in the design of gas turbine blades. However, the underlying mathematical coupling structures have not been thoroughly investigated. In this work, the port-Hamiltonian formalism is applied to the conjugate heat transfer problem in gas turbine blades. A mathematical modelbbased on common engineering simplifications is constructed and further simplified to reduce complexity and focus on the coupling structures of interest. The model is then cast as a port-Hamiltonian system and examined for stability and well-posedness.
Krylov Subspace Recycling for Evolving Structures
Authors: Matthias Bolten, Eric de Sturler, Camilla Hahn
Abstract: Krylov subspace recycling is a powerful tool for solving long series of large, sparse linear systems that change slowly. In PDE constrained shape optimization, these appear naturally, as hundreds or more optimization steps are needed with only small changes in the geometry. In this setting, however, applying Krylov subspace recycling can be difficult. As the geometry evolves, so does the finite element mesh, especially if re-meshing is needed. As a result, the number of algebraic degrees of freedom in the system may change from one optimization step to the next, and with it the size of the finite element system matrix. Changes in the mesh also lead to structural changes in the matrices. In the case of remeshing, even if the geometry changes only a little, the corresponding mesh might differ substantially from the previous one. read moreThis prevents any straightforward mapping of the approximate invariant subspace of the linear system matrix (the focus of recycling in this paper) from one step to the next; similar problems arise for other selected subspaces. We present an algorithm for general meshes to map an approximate invariant subspace of the system matrix for the previous optimization step to an approximate invariant subspace of the system matrix for the current optimization step. We exploit the map from coefficient vectors to finite element functions on the mesh combined with function approximation on the finite element mesh. In addition, we develop a straightforward warm-start adaptation of the Krylov-Schur algorithm [G.W. Stewart, SIAM J. Matrix Anal. Appl. 23, 2001] to improve the approximate invariant subspace at the start of a new optimization step if needed. We demonstrate the effectiveness of our approach numerically with several proof of concept studies for a specific meshing technique.
Hypervolume Scalarization for Shape Optimization to Improve Reliability and Cost of Ceramic Components
Authors: Camilla Hahn, Kathrin Klamroth, Johanna Schultes and Michael Stiglmayr
Abstract: In engineering applications one often has to trade-off among several objectives as, for example, the mechanical stability of a component, its efficiency, its weight and its cost. We consider a biobjective shape optimization problem maximizing the mechanical stability of a ceramic component under tensile load while minimizing its volume. Stability is thereby modeled using a Weibull-type formulation of the probability of failure under external loads. The PDE formulation of the mechanical state equation is discretized by a finite element method on a regular grid. To solve the discretized biobjective shape optimization problem we suggest a hypervolume scalarization, with which also unsupported efficient read moresolutions can be determined without adding constraints to the problem formulation. We investigate the relation of the hypervolume scalarization to the weighted sum scalarization and to direct multiobjective descent methods. Since gradient information can be efficiently obtained by solving the adjoint equation, the scalarized problem can be solved by a gradient ascent algorithm. We evaluate our approach on a 2 D test case representing a straight joint under tensile load.
An Analytical Study in Multi Physics and Multi Criteria Shape Optimization
Authors: Hanno Gottschalk and Marco Reese
Abstract: A simple multi-physical system for the potential flow of a fluid through a shroud in which a mechanical component, like a turbine vane, is placed, is modeled mathematically. We then consider a multi criteria shape optimization problem, when the shape of the component is allowed to vary under a certain set of 2nd order Hölder continuous differentiable transformations of a baseline shape with boundary of the same continuity class. As objective functions, we consider a simple loss model for the fluid dynamical efficiency and the probability of failure of the component due to repeated application of loads that stem from the fluid’s static pressure. read moreFor this multi-physical system, it is shown that under certain conditions the Pareto front is maximal in the sense that the Pareto front of the feasible set coincides with Pareto front of its closure. We also show that the set of all optimal forms with respect to scalarization techniques deforms continuously (in the Hausdorff metric) with respect to preference parameters.
Tracing Locally Pareto Optimal Points by Numerical Integration
Authors: Matthias Bolten, Onur Tanil Doganay, Hanno Gottschalk and Kathrin Klamroth
Abstract: We suggest a novel approach for the efficient and reliable approximation of the Pareto front of sufficiently smooth unconstrained bi-criteria optimization problems. Optimality conditions formulated for weighted sum scalarizations of the problem yield a description of (parts of) the Pareto front as a parametric curve, parameterized by the scalarization parameter (i.e., the weight in the weighted sum scalarization). Its sensitivity w.r.t. parameter variations can be described by an ordinary differential equation (ODE). Starting from an arbitrary initial Pareto optimal solution, the Pareto front can then be traced by numerical integration. read moreWe provide an error analysis based on Lipschitz properties and suggest an explicit Runge-Kutta method for the numerical solution of the ODE. The method is validated on bi-criteria convex quadratic programming problems for which the exact solution is explicitly known, and numerically tested on complex bi-criteria shape optimization problems involving finite element discretizations of the state equation
Towards Multidisciplinary Turbine Blade Tolerance Design Assessment using Adjoint Methods
Authors: Alexander Liefke and Peter Jaksch and Sebastian Schmitz and Vincent Marciniak
Abstract: This paper shows how to use discrete CFD and FEM ad-joint surface sensitivities to derive objective-based tolerances for turbine blades, instead of relying on geometric tolerances. For this purpose a multidisciplinary adjoint evaluation tool chain is introduced to quantify the effect of real manufacturing imperfections on aerodynamic efficiency and probabilistic low cycle fatigue life time. Before the adjoint method is applied, a numerical validation of the CFD and FEM adjoint gradients is performed using 102 heavy duty turbine vane scans. The results show that the absolute error for adjoint CFD gradients is below 0.5%, read morewhile the FEM life time gradient absolute errors are below 5%. The adjoint assessment tool chain further reduces the computational cost by around 85% for the investigated test case compared to non-linear methods. Through the application of the presented tool chain, the definition of specified objective-based tolerances becomes available as a design assessment tool and allows to improve overall turbine efficiency and the accuracy of life time prediction.
GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks (Preprint)
Authors: Backhaus, J. and Bolten, M. and Doganay, O.T., and Ehrhardt, M. and Engel, B. and Frey, Ch. and Gottschalk, H. and Günther, M. and Hahn, C., Jaschke, J., and Jaksch, P. and Klamroth, K. and Liefke, A. and Luft, D. Mäde, L. and Marciniak, V. and Reese, M. and Schultes, J. and Schulz, V. and Schmitz, S. and Steiner, J. and Stiglmayr, M.
Abstract: This paper describes the project GivEn that develops a novel multicrite-ria optimization process for gas turbine blades and vanes using modern ”adjoint”shape optimization algorithms. read moreGiven the many start and shut-down processes ofgas power plants in volatile energy grids, besides optimizing gas turbine geometriesfor efficiency, the durability understood as minimization of the probability of fail-ure is a design objective of increasing importance. We also describe the underlyingcoupling structure of the multiphysical simulations and use modern, gradient based multicriteria optimization procedures to enhance the exploration of Pareto-optimalsolutions.
Probabilistic Modeling of Slip System-Based Shear Stresses and Fatigue Behavior of Coarse-Grained Ni-Base Superalloy Considering Local Grain Anisotropy and Grain Orientation
Authors: Benedikt Engel 1, Lucas Mäde, Philipp Lion, Nadine Moch, Hanno Gottschalk and Tilmann Beck
Abstract: New probabilistic lifetime approaches for coarse grained Ni-base superalloys supplement current deterministic gas turbine component design philosophies; in order to reduce safety factors and push design limits. The models are based on statistical distributions of parameters, which determine the fatigue behavior under high temperature conditions. In the following paper, Low Cycle Fatigue (LCF) test data of several material batches of polycrystalline Ni-base superalloy René80 with different grain sizes and orientation distribution (random and textured) is presented and evaluated. read moreThe textured batch, i.e., with preferential grain orientation, showed higher LCF life. Three approaches to probabilistic crack initiation life modeling are presented. One is based on Weibull distributed crack initiation life while the other two approaches are based on probabilistic Schmid factors. In order to create a realistic Schmid factor distribution, polycrystalline finite element models of the specimens were generated using Voronoi tessellations and the local mechanical behavior investigated in dependence of different grain sizes and statistically distributed grain orientations. All models were first calibrated with test data of the material with random grain orientation and then used to predict the LCF life of the material with preferential grain orientation. By considering the local multiaxiality and resulting inhomogeneous shear stress distributions, as well as grain interaction through polycrystalline Finite Element Analysis (FEA) simulation, the best consistencies between predicted and observed crack initiation lives could be achieved.
Gradient Based Biobjective Shape Optimization to Improve Reliability and Cost of Ceramic Components
Authors: Onur T. Doganay, Camilla Hahn, Hanno Gottschalk, Kathrin Klamroth, Johanna Schultes, Michael Stiglmayr
Abstract: We consider the simultaneous optimization of the reliability and the cost of a ceramic component in a biobjective PDE constrained shape optimization problem. A probabilistic Weibull-type model is used to assess the probability of failure of the component under tensile load, while the cost is assumed to be proportional to the volume of the component. Two different gradient-based optimization methods are suggested and compared at 2D test cases. read moreThe numerical implementation is based on a first discretize then optimize strategy and benefits from efficient gradient computations using adjoint equations. The resulting approximations of the Pareto front nicely exhibit the trade-off between reliability and cost and give rise to innovative shapes that compromise between these conflicting objectives.
- SIAM Journal on Optimization
Efficient Techniques for Shape Optimization with Variational Inequalities using Adjoints
Authors: Daniel Luft, Volker H. Schulz, Kathrin Welker
Abstract: In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for semi-smooth shape optimization problems. In this paper, we consider shape optimization problems constrained by variational inequalities of the first kind, so-called obstacle-type problems. Under appropriate assumptions, we prove existence of adjoints for regularized problems and convergence…read moreto adjoints of the unregularized problem. Moreover, we derive shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically.
Shape optimization to decrease failure probability
Authors: Matthias Bolten, Hanno Gottschalk, Camilla Hahn, Mohamed Saadi
Abstract: Ceramic is a material frequently used in industry because of its favorable properties. Common approaches in shape optimization for ceramic structures aim to minimize the tensile stress acting on the component, as it is the main driver for failure. In contrast to this, we follow a more natural approach by minimizing the component’s probability of failure under a given tensile load. Since the…read morefundamental work of Weibull, the probabilistic description of the strength of ceramics is standard and has been widely applied. Here, for the first time, the resulting failure probabilities are used as objective functions in PDE constrained shape optimization. To minimize the probability of failure, we choose a gradient based method combined with a first discretize then optimize approach. For discretization finite elements are used. Using the Lagrangian formalism, the shape gradient via the adjoint equation is calculated at low computational cost. The implementation is verified by comparison of it with a finite difference method applied to a minimal 2d example. Furthermore, we construct shape flows towards an optimal / improved shape in the case of a simple beam and a bended joint.
Shape gradients for the failure probability of a mechanical component under cyclical loading
Authors: Hanno Gottschalk, Mohamed Saadi
Abstract: This work provides a numerical calculation of shape gradients of failure probabilities for mechanical components using a first discretize, then adjoint approach. While deterministic life prediction models for failure mechanisms are not (shape) differentiable, this changes in the case of probabilistic life prediction. The probabilistic, or reliability based, approach thus opens the way for…read moreefficient adjoint methods in the design for mechanical integrity. In this work we propose, implement and validate a method for the numerical calculation of the shape gradients of failure probabilities for the failure mechanism low cycle fatigue (LCF), which applies to polycrystalline metal. Numerical examples range from a bended rod to a complex geometry from a turbo charger in 3D.
Adjoint Method to Calculate Shape Gradients of Failure Probabilaties for Turbomachinery Components
Authors: Hanno Gottschalk, Mohamed Saadi, Onur Tanil Doganay, Kathrin Klamroth, Sebastian Schmitz
Abstract: In the optimization of turbomachinery components, shape sensitivities for fluid dynamical objective functions have been used for a long time. As peak stress is not a differential functional of the shape, such highly efficient procedures so far have been missing for objective functionals that stem from mechanical integrity. This changes, if deterministic lifing criteria are replaced by…read moreprobabilistic criteria, which have been introduced recently to the field of low cycle fatigue (LCF). Here we present a finite element (FEA) based first discretize, then adjoin approach to the calculation of shape gradients (sen- sitivities) for the failure probability with regard to probabilistic LCF and apply it to simple and complex geometries, as e.g. a blisk geometry. We review the computation of failure probabilities with a FEA postprocessor and sketch the computation of the relevant quantities for the adjoint method. We demonstrate high accuracy and computational efficiency of the adjoint method compared to finite difference schemes. We discuss implementation details for rotating components with cyclic boundary conditions. Finally, we shortly comment on future development steps and on potential applications in multi criteria optimization.
Probabilistic LCF Risk Evaluation of a Turbine Vane by Combined Size Effect and Notch Support Modeling
Authors: Lucas Mäde, Sebastian Schmitz, Hanno Gottschalk, Tilman Beck
Abstract: A probabilistic risk assessment for low cycle fatigue (LCF) based on the so-called size effect has been applied on gas-turbine design in recent years. In contrast, notch support modeling for LCF which intends to consider the change in stress below the surface of critical LCF regions is known and applied for decades. Turbomachinery components often show sharp stress gradients and very localized…read morecritical regions for LCF crack initiations so that a life prediction should also consider notch and size effects. The basic concept of a combined probabilistic model that includes both, size effect and notch support, is presented. In many cases it can improve LCF life predictions significantly, in particular compared to curve predictions of standard specimens where no notch support and size effect is considered. Here, an application of such a combined model is shown for a turbine vane.
- Conference: ECCM-ECFD 2018, At Glasgow
Using Adjoint CFD to Quantify the Impact of Manufacturing Variations on a Heavy Duty Turbine VaneAuthors: Alexander Liefke, Vincent Marciniak, Uwe Janoske, Hanno Gottschalk
Abstract: Turbine efficiency is one of the main design criteria for heavy duty gas turbines. After the design, margin adaption factors are applied on the baseline to predict the impact of manufacturing variations (MV). These margins are normally based on testbed experience. A more detailed knowledge of the impact of MV, prior to testing, would therefore improve the margin prediction accuracy and could benefit in product cost and global efficiency. …read more
For turbomachines the impact of MV can be quantified with a Monte Carlo (MC) simulation in combination with steady non-linear CFD calculations e.g. RANS. The drawback of this approach is the large number of RANS computations needed to quantify the impact of MV, which is prohibitive for a daily use in an industrial context. Assuming that the MV are small enough, the adjoint CFD method, which linearizes the governing equations, can be an alternative to the RANS evaluations. This kind of approach has been successfully used for compressors and turbines.
The first part of this paper presents a systematic approach to evaluate a hand-derived and an algorithmic-derived version of the discrete adjoint CFD solver TRACE. To do so, the ERCOFTAC axial flow turbine known as Aachen Turbine has been selected. For the adjoint version comparison a NACA-like parametrization is applied to compare and validate the adjoint-generated with finite difference gradients.
In the second part the adjoint-based method is applied to an industrial turbine vane to quantify the impact of MV. For this case real MV have been measured using 102 optical blade scans. The scans are used to generate the corresponding deformed geometries for which an adjoint and a RANS simulation are computed. The comparison between each computation demonstrates that the impact of realistic MV can be handled by the adjoint approach.