Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains

In this work we are interested in the numerical solution of the steady incompressible Navier-Stokes equations for fluid flow in pipes with varying curvatures and cross-sections. We intend to compute a reduced basis approximation of the solution, employing the geometry as a parameter in the reduced basis method. This has previously been done in a spectral element $P_{{ \mathcal{N}}} - P_{{ \mathcal{N}}-2}$ setting in two dimensions for the steady Stokes equations. To compute the necessary basis-functions in the reduced basis method, we propose to use a stabilized P 1−P 1 finite element method for solving the Navier-Stokes equations on different geometries. By employing the same stabilization in the reduced basis approximation, we avoid having to enrich the velocity basis in order to satisfy the inf-sup condition. This reduces the complexity of the reduced basis method for the Navier-Stokes problem, while keeping its good approximation properties. We prove the well posedness of the reduced problem and present numerical results for selected parameter dependent three dimensional pipe

Reduced Basis Method for the Stokes Equations in Decomposable Parametrized Domains Using Greedy Optimization

Flow simulations in pipelined channels and several kinds of parametrized configurations have a growing interest in many life sciences and industrial applications. Applications may be found in the analysis of the blood flow in specific compartments of the circulatory system that can be represented as a combination of few deformed vessels from reference ones, e.g. pipes. We propose a solution approach that is particularly suitable for the study of internal flows in hierarchical parametrized geometries. The main motivation is for applications requiring rapid and reliable numerical simulations of problems in domains involving parametrized complex geometries. The classical reduced basis (RB) method is very effective to address viscous flows equations in parametrized geometries (see, e.g., [10]). An interesting alternative foresees a combination of RB with a domain decomposition approach. In this respect, preliminary efforts to reduce the global parametrized problem to local ones have led to the introduction of the so-called reduced basis element method to solve the Stokes problem [6], and more recently to the reduced basis hybrid method [3] and to the static condensation method [7]. In general, we are interested in defining a method able to maintain the flexibility of dealing with arbitrary combinations of subdomains and several geometrical deformations of the latter. A further new contribution to this field is the computation of the reduced basis functions through an optimization greedy algorithm

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references

How therapists in cognitive behavioral and psychodynamic therapy reflect upon the use of metaphors in therapy: a qualitative study

Background Research suggests that metaphors are integral to psychotherapeutic practice. We wanted to explore how 10 therapists reflect upon the use of metaphors in therapy, and how they react to some metaphors expressed by patients treated for of major depressive disorder (MDD). Methods Five therapists practicing psychodynamic therapy (PDT) and five practicing cognitive behavioral therapy (CBT) were interviewed with a semi-structured qualitative interview. Transcripts were analyzed using a thematic analysis approach. Results Our analysis resulted in two main themes: the therapeutic use of metaphors, and conflicting feelings towards metaphors used by depressed patients. Most therapists said that they do not actively listen for metaphors in therapy and many said that they seldom use metaphors deliberately. While PDT-therapists appeared more attentive to patient-generated metaphors, CBT-therapists seemed more focused on therapist-generated metaphors. Most therapists did not try to alter the patient-generated metaphors they evaluated as unhelpful or harmful. Some therapists expressed strong negative feelings towards some of the metaphors used by patients. PDT-therapists were the most critical towards the metaphor of tools and the metaphor of depression as an opponent. CBT-therapists were the most critical towards the metaphor of surface-and-depth. Conclusions These results remind us of the complexity of using metaphors in therapy, and can hopefully be an inspiration for therapists to reflect upon their own use of metaphors. Open therapeutic dialogue on the metaphor of tools, surface-depth and depression as an opponent may be necessary to avoid patient-therapist-conflicts. Trial registration Clinical Trial gov. Identifier: NCT03022071 . Date of registration: 16/01/2017

Digging down or scratching the surface: how patients use metaphors to describe their experiences of psychotherapy

Background In the present study, we wanted to explore which metaphors patients suffering from major depressive disorder (MDD) use to explain their experience of being in therapy and their improvement from depression. Methods Patients with MDD (N = 22) received either psychodynamic therapy (PDT) or cognitive behavioral therapy (CBT). They were interviewed with semi-structured qualitative interviews after ending therapy. The transcripts were analyzed using a method based on metaphor-led discourse analysis. Results Metaphors were organized into three different categories concerning the process of therapy, the therapeutic relationship and of improvement from depression. Most frequent were the metaphorical concepts of surface and depth, being open and closed, chemistry, tools, improvement as a journey from darkness to light and depression as a disease or opponent. Conclusions Patient metaphors concerning the therapeutic experience may provide clinicians and researchers valuable information about the process of therapy. Metaphors offer an opportunity for patients to communicate nuances about their therapeutic experience that are difficult to express in literal language. However, if not sufficiently explored and understood, metaphors may be misinterpreted and become a barrier for therapeutic change. Trial registration Clinical Trial gov. Identifier: NCT03022071. Date of registration: 16/01/2017

A REDUCED BASIS ELEMENT METHOD FOR COMPLEX FLOW SYSTEMS

Key words: reduced basis, domain decomposition, a posteriori error estimators, transfinite interpolation, empirical interpolation Abstract. The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations within domains belonging to a certain class. The method takes its roots in domain decomposition methods and reduced basis discretizations. 1–3 The basic idea is to first decompose the computational domain into smaller blocks that are topologically similar to a few reference shapes (or generic computational parts). Associated with each reference shape are precomputed solutions corresponding to the same governing partial differential equation, and similar boundary conditions, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. 4,5 The approximation corresponding to the computational domain is then taken to be a linear combination of the precomputed solutions, mapped from the reference shapes for the different blocks to the actual domain. The variation of the geometry induces non-affine parameter dependence, and we apply the empirical interpolation technique to achieve a