The topological derivative method for optimum shape design and control of gas networks
Bibliographic record
Abstract
In this paper, topological derivatives are defined and employed for gas transport networks governed by nonlinear hyperbolic systems of PDEs.The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks.First, the dynamic model for the network is considered.The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes.For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified.That is, the design of the network can be performed for optimal control of the steady-state network model.The cost of design is defined by the optimal control cost for the steady-state network model.The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is determined.Tree-structured networks can be decomposed into single network junctions.The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network.This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost.As the set of network junctions is finite, the iterative procedure is convergent.This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs.A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".