The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and ...use continuous solutions, unlike previous efforts that use discontinuous SBP discretizations. Derivative-based dissipation and local-projection stabilization (LPS) are investigated as options for stabilizing the baseline discretization. These stabilizations are equal up to a multiplicative constant in one dimension, but only LPS remains well conditioned for general, multidimensional SBP operators. Furthermore, LPS is able to take advantage of the additional nodes required by degree 2
p
diagonal-norms, resulting in an element-local stabilization with a bounded spectral radius. An entropy-stable version of LPS is easily obtained by applying the projection on the entropy variables. Numerical experiments with the linear-advection and Euler equations demonstrate the accuracy, efficiency, and robustness of the stabilized discretizations, and the continuous approach compares favorably with the more common discontinuous SBP methods.
Improved constraint-aggregation methods Kennedy, Graeme J.; Hicken, Jason E.
Computer methods in applied mechanics and engineering,
06/2015, Letnik:
289
Journal Article
Recenzirano
Many constraints in design-optimization problems take the form of bounds on a physical quantity over a domain. These infinite-dimensional constraints are often addressed through aggregation methods ...that approximate the bound in a differentiable manner. However, the most common constraint-aggregation methods exhibit mesh dependence; as the mesh spacing decreases, the aggregation function diverges. In addition, there are no established methods to assess the impact of constraint aggregation on the optimized design. To address these deficiencies, we propose a new class of constraint-aggregation method that we call induced aggregates. We examine the properties of these aggregation methods, describe their numerical implementation, and test their numerical accuracy on a pressure-loaded circular cylinder test case. Furthermore, we propose a post-optimality estimation technique to assess the impact of the constraint-aggregation method on the design. We use this method to examine the optimized mass obtained from two stress-constrained mass minimization problems: a variable-thickness sheet problem and a wingbox design problem.
•We develop constraint-aggregation methods for bound-constrained design optimization.•We introduce induced aggregates: a new class of constraint-aggregation method.•We examine the properties of induced aggregates and their numerical implementation.•We develop a post-optimality technique to assess the impact of constraint-aggregation.•We apply these techniques to two stress-constrained mass-minimization problems.
•History of SBP operators and SATs, with a focus on the Navier–Stokes (NS) equations.•Methodology for deriving SBP operators for 1st and 2nd derivatives.•Implementation of SBP-SATs for the numerical ...solution of the NS equations.•Review of recent developments, including functional superconvergence.
Summation-by-parts (SBP) operators have a number of properties that make them an attractive option for higher-order spatial discretizations of partial differential equations. In particular, they enable the derivation of higher-order boundary closures leading to provable time stability. When implemented on multi-block structured meshes in conjunction with simultaneous approximation terms (SATs)—penalty terms that impose boundary and interblock-coupling conditions in a weak sense—they offer additional properties of value, even for second-order accurate schemes and steady problems. For example, they involve low communication overhead for efficient parallel algorithms and relax the continuity requirements of both the mesh and the solution across block interfaces. This paper provides a brief history of seminal contributions to, and applications of, SBP-SAT methods followed by a description of their properties and a methodology for deriving SBP operators for first derivatives and second derivatives with variable coefficients. A procedure for deriving SATs is also provided. Practical aspects are discussed, including artificial dissipation, transformation to curvilinear coordinates, and application to the Navier–Stokes equations. Recent developments are reviewed, including a variational interpretation, the connection to quadrature rules, functional superconvergence, error estimates, and dual consistency. Finally, the connection to quadrature rules is exploited to provide a generalization of the SBP concept to a broader class of operators, enabling a unification and rigorous development of SATs for operators such as nodal-based pseudo-spectral and some discontinuous Galerkin operators.
We present and analyze an entropy-stable semi-discretization of the Euler equations based on high-order summation-by-parts (SBP) operators. In particular, we consider general multidimensional SBP ...elements, building on and generalizing previous work with tensor–product discretizations. In the absence of dissipation, we prove that the semi-discrete scheme conserves entropy; significantly, this proof of nonlinear L2 stability does not rely on integral exactness. Furthermore, interior penalties can be incorporated into the discretization to ensure that the total (mathematical) entropy decreases monotonically, producing an entropy-stable scheme. SBP discretizations with curved elements remain accurate, conservative, and entropy stable provided the mapping Jacobian satisfies the discrete metric invariants; polynomial mappings at most one degree higher than the SBP operators automatically satisfy the metric invariants in two dimensions. In three-dimensions, we describe an elementwise optimization that leads to suitable Jacobians in the case of polynomial mappings. The properties of the semi-discrete scheme are verified and investigated using numerical experiments.
•Entropy-stable, SBP discretizations for non-tensor product, curved elements.•Polynomial mappings can be at most one degree higher than the SBP operators.•3D mapping Jacobian found by solving an elementwise optimization problem.•Numerical experiments verify entropy stability and accuracy.
This paper presents a novel multi-fidelity model-management framework based on the estimated error between the low-fidelity and high-fidelity models. The optimization algorithm is similar to ...classical multi-fidelity trust-region model-management approaches, but it replaces the trust-radius constraint with a bound on the estimated error between the low- and high-fidelity models. This enables globalization without requiring the user to specify non-intuitive parameters such as the initial trust radius, which have a significant impact on the cost of the optimization yet can be hard to determine a priori. We demonstrate the framework on a simple one-dimensional optimization problem, a series of analytical benchmark problems, and a realistic electric-motor optimization. We show that for low-fidelity models that accurately capture the trends of the high-fidelity model, the developed framework can significantly improve the efficiency of obtaining high-fidelity optima compared to state-of-the-art multi-fidelity optimization methods and a direct high-fidelity optimization.
We present a gradient-enhanced, partition-of-unity surrogate model and an associated adaptive sampling procedure that performs well for functions with low regularity or rapid variation over short ...distances. The approach reconstructs local quadratic models at each training point using function and gradient data from neighboring points, and partition-of-unity basis functions blend these local models. We apply the proposed method to a suite of analytical test functions and a coupled aerostructural model to demonstrate its relative effectiveness. The performance of the partition-of-unity surrogate is assessed relative to Kriging and gradient-enhanced Kriging (GEK) models constructed from appropriate one-shot and adaptive sampling methods. The proposed model obtains results competitive with those of the equivalent Kriging models for the tested functions, and it notably outperforms Kriging for functions with slowly decaying spectra, such as those featuring strong jumps, non-smoothness, and low regularity. Furthermore, the model achieves its competitive accuracy without requiring hyperparameter training.
We formulate the immersed‐boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is ...the one that minimizes the mismatch between the state and the desired boundary value along the immersed target‐domain boundary. We begin by investigating a naïve problem formulation that we show is ill‐posed: in the case of the Laplace equation, we prove that the solution is unique, but it fails to depend continuously on the data; for the linear advection equation, even solution uniqueness fails to hold. These issues are addressed by two complimentary strategies. The first strategy is to ensure that the enclosing domain tends to the true domain, as the mesh is refined. The second strategy is to include a specialized parameter‐free regularization that is based on penalizing the difference between the control and the state on the boundary. The proposed inverse IBM is applied to the diffusion, advection, and advection‐diffusion equations using a high‐order discontinuous Galerkin discretization. The numerical experiments demonstrate that the regularized scheme achieves optimal rates of convergence and that the reduced Hessian of the optimization problem has a bounded condition number, as the mesh is refined.
We formulate the immersed‐boundary method as an inverse problem, in which a control variable is introduced on the boundary of the computational domain to minimize the mismatch between the state and the desired boundary value. The inverse problem is stabilized by including a specialized parameter‐free regularization. The method is applied to the diffusion, advection, and advection‐diffusion equations, and the numerical experiments show optimal accuracy and bounded condition number of the reduced Hessian of the optimization problem.
The entropy conservative/stable staggered grid tensor-product algorithm of Parsani et al. 1 is extended to multidimensional SBP discretizations. The required SBP preserving interpolation operators ...are proven to exist under mild restrictions and the resulting algorithm is proven to be entropy conservative/stable as well as elementwise conservative. For 2-dimensional simplex elements, the staggered grid algorithm is shown to be more accurate and have a larger maximum time step restriction as compared to the collocated algorithm. The staggered algorithm significantly reduces the number of (computationally expensive) two-point flux evaluations, which is potentially important for both explicit and implicit time-marching schemes. Furthermore, the staggered algorithm requires fewer degrees of freedom for comparable accuracy, which has favorable implications for implicit time-marching schemes.
•Develop an entropy conservative/stable scheme for multi-D SBP operators.•Prove that prolongation/restriction operator pairs exist under mild assumptions.•The staggered algorithm is more accurate and reduces the DOFs.•The staggered algorithm allows a larger stable time step.
This work focuses on simultaneous approximation terms (SATs) for multidimensional summation-by-parts (SBP) discretizations of linear second-order partial differential equations with variable ...coefficients. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties for general operators, including those operators that do not have nodes on their boundary or do not correspond with a collocation discontinuous Galerkin method. Based on these conditions, we generalize the modified scheme of Bassi and Rebay and the symmetric interior penalty Galerkin method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.
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