Artykuły w czasopismach na temat „Spatial H-2 norm”

Abstract We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming $hp$-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for $L^2(H^1)$-norm estimates. Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the error plus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim \tau $. This result improves on earlier works that required stronger two-sided hypotheses such as $h \simeq \tau $ or $h^2\simeq \tau $; instead, our result now encompasses practically relevant cases for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees and the refinement and coarsening between time steps, thereby removing any transition condition.

In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order O ( τ 2 + h 4 ) (where τ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also O ( τ 2 + h 4 ) under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis.

AbstractA robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the {P_{1}}-continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex–Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. {L^{2}} norm) is bounded by {O(h+k)} (resp. {O(h^{2}+k)}), where h and k denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lamé coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison.

Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.

<abstract><p>In this paper, we are concerned with optimal decay rates of higher–order derivatives of the smooth solutions to the $ 3D $ compressible nematic liquid crystal flows. The main novelty of this paper is three–fold: First, under the assumptions that the initial perturbation is small in $ H^N $–norm $ (N\geq3) $ and bounded in $ L^1 $–norm, we show that the highest–order spatial derivatives of density and velocity converge to zero at the $ L^2 $–rates is $ (1+t)^{-\frac{3}{4}-\frac{N }{2 }} $, which are the same as ones of the heat equation, and particularly faster than the $ L^2 $–rate $ (1+t)^{-\frac{1}{4}-\frac{N }{2 }} $ in [J.C. Gao, et al., J. Differential Equations, 261: 2334-2383, 2016]. Second, if the initial data satisfies some additional low frequency assumption, we also establish the lower optimal decay rates of solution as well as its all–order spatial derivatives. Therefore, our decay rates are optimal in this sense. Third, we prove that the lower bound of the time derivatives of density, velocity and macroscopic average converge to zero at the $ L^2 $–rate is $ (1+t)^{-\frac{5}{4}} $. Our method is based on low-frequency and high-frequency decomposition and energy methods.</p></abstract>

ABSTRACT The spatial distribution between dark matter and baryonic matter of the Universe is biased or deviates from each other. In this work, by comparing the results derived from IllustrisTNG and WIGEON simulations, we find that many results obtained from TNG are similar to those from WIGEON data, but differences between the two simulations do exist. For the ratio of density power spectrum between dark matter and baryonic matter, as scales become smaller and smaller, the power spectra for baryons are increasingly suppressed for WIGEON simulations; while for TNG simulations, the suppression stops at $k=15-20\, {h {\rm Mpc}^{-1}}$, and the power spectrum ratios increase when $k\gt 20\, {h {\rm Mpc}^{-1}}$. The suppression of power ratio for WIGEON is also redshift-dependent. From z = 1 to z = 0, the power ratio decreases from about 70 per cent to less than 50 per cent at $k=8\, {h {\rm Mpc}^{-1}}$. For TNG simulation, the suppression of power ratio is enhanced with decreasing redshifts in the scale range $k\gt 4\, {h {\rm Mpc}^{-1}}$, but is nearly unchanged with redshifts in $k\lt 4\, {h {\rm Mpc}^{-1}}$. These results indicate that turbulent heating can also have the consequence to suppress the power ratio between baryons and dark matter. Regarding the power suppression for TNG simulations as the norm, the power suppression by turbulence for WIGEON simulations is roughly estimated to be 45 per cent at $k=2\, {h {\rm Mpc}^{-1}}$, and gradually increases to 69 per cent at $k=8\, {h {\rm Mpc}^{-1}}$, indicating the impact of turbulence on the cosmic baryons are more significant on small scales.

Abstract In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 {\mathbf{H}_{0}^{1}} . New regularity results which are valid uniformly in time as t → ∞ {t\to\infty} and in the penalty parameter ε as ε → 0 {\varepsilon\to 0} are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 {\mathbf{L}^{2}} as well in 𝐇 1 {\mathbf{H}^{1}} -norms and for the penalized pressure in L 2 {L^{2}} -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.

<p style="text-indent:20px;">This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the <inline-formula><tex-math id="M1">\begin{document}$ L^2\cap L^\infty- $\end{document}</tex-math></inline-formula>norm of the initial perturbation is assumed to be small, the <inline-formula><tex-math id="M2">\begin{document}$ H^1- $\end{document}</tex-math></inline-formula>norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy method.</p>

Abstract In this paper, we study well-posedness, existence of a lower finite time blow-up bound and variants of controllability of the classical chemotaxis model in {\Omega\times(0,T)} , where {\Omega\subset\mathbb{R}^{N}} , {N=1,2,3} . The spatial domain restrictions allow the system with initial data in {L^{2}(\Omega)} to admit a solution in L^{\infty}[0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega)) and to have the property that the gradient chemical solutions are uniformly bounded in {\Omega\times(0,T)} . A lower finite time blow-up bound of solutions in the norm of {L^{2}(\Omega)} is proved using the differential inequality technique. Furthermore, using Carleman estimates and appropriate energy functionals, we show that the model is null and approximate controllable at any finite time {T>0} with a single control in {L^{2}(\omega\times(0,T))} acting on the cell-density equation, linearized through a priori uniform boundedness of the chemical drift solutions, where {\omega\subset\Omega} is a non-empty open subset of Ω. Lastly, bang-bang-type controls for the problem are constructed.

The GIS-ENDDB expert-information system includes the global seismic catalogs and measurement data for various geophysical fields. Statistical processing of these data and visualization of the distributing characteristics of the seismic-geodynamic process in the form of isolines make it possible to reveal the spatial relationship between the seismicity distribution and anomalies of geophysical fields. The first attempts to identify this relationship were made with L.S. Sokolova in 2016 based on a visual comparison of maps obtained as a result of applying methods of various data analysis (in particular, the isolines maps of the maximum magnitude of recorded earth-quakes Mmax) for different samples of seismological catalogs on the one hand, and various modifications of the measured geophysical fields (in this case, the regional component of heat flow (HF) field) on the other hand. These results were more likely qualitative, but later a correlation analysis function was implemented into GIS-ENDDB for the quantitative comparison of the fields. The correlation analysis is carried out on a histogram of the distributing number of cells (nodes) NC from the difference ∆F of the reduced values of the fields F1 and F2 (relative to the arithmetic mean Fi.OCP) in the cells that are equal for the compared fields. To identify non-random variations, NC histograms (∆F) are given against the background of the normal distribution curve, which allows not only visually (by the deviation of the first columns from the norm), but also quantitatively to establish the degree of correlation. For the studied territories, histograms have obtained that show: 1. a moderate correlation of HF and Mmax fields in the North American continent (NEIC catalog, M≥5, cell 3×4); 2. no correlation across Asia - Far East region (COMPLEX, M≥4, H≥50 km, cell 6×8); and 3. high correlation in a rectangular region covering the Inside-Asian mountain belt (30-40N; 66-84E., COMPLEX, M≥5, H≥50 km, cell 1×1).

In July, 1992, fire swept through the Jelonka Reserve (52^o35'33"N; 23^o22'10"E). Litter, herbs layer and juniper shrubs were razed and the tree trunks of <em>Pinus sylvestris</em> and <em>Populus tremula</em> damaged, which resulted in their death the following year. Five research plots of 25 m² each were established in the post-fire area. The species present there and their degrees of coverage were listed on the Londo scale between 1993 and 2002. The calculation of average degrees of coverage for individual populations allowed for the establishing of a chronological pattern of post-fire succession. The following findings have been made: 1) the primary function in colonization belongs to species of guerilla strategy of growth whose dormant buds survived fire underground (<em>Holcus mollis</em>, <em>Calluna vulgaris</em> and <em>Populus tremula</em>) or phalanx strategy of growth (<em>Corynephoms canescens</em>); 2) the dominants of the initial succession stage are perennial grasses, not therophytes; 3) the spatial pattern of succession is influenced by the occurrence of <em>H. mollis</em> or <em>C. canescens</em> in a particular location; 4) the initial stage terminates with the occurrence and further growth of <em>C. vulgaris</em> clumps; the post-fire succession in the Jelonka Reserve attained a brushwood stage faster in comparison to secondary succession, formerly triggered by the cessation of cultivation on arable land.

To detect the cyclic component in time series of annual water runoff of rivers of the Siverskyi Donets River Basin structure, it is necessary to have a hydrological gauge, which closes a large river basin and has long continuous observations of water runoff. The only hydrological gauge that meets these conditions is the Siverskyi Donets – Lysychansk – the catchment area is 52,400 km2 and the beginning of observations of water runoff since 1892. The Siverskyi Donets – Lysychansk is quite intermittent. Analysis of annual water runoff data of “neighboring basins” for the Siverskyi Donets Basin showed that the longest series of continuous observations has a hydrological gauge the Desna River near Chernihiv – since 1895, the catchment area is 81400 km2. So, we can consider this basin as basic for calculations and determination of patterns of long-term variability of annual water runoff of rivers of the Siverskyi Donets Basin. The study used data from eight hydrological gauges: the Desna – Chernihiv, intermediate river basins – the Sula – Lubny, the Psel – Zapsillya, the Vorskla – Kobeliaky, within the study basin – the Siverskyi Donets – Lysychansk, the Bakhmut – Siversk, the Aidar – Novoselivka, as well as the Southern Bug – Oleksandrivka. To confirm the spatial consistency of the annual water runoff of the studied rivers, a correlation matrix between the time sequences of the water runoff of neighboring basins was determined. To obtain a generalized characteristic of long-term water runoff variability of the studied rivers, chronological graphs of changes in modular coefficients were constructed, initial time series smoothing was performed, graphical analysis of difference integral curves was applied. In the course of the research regularities in long-term variability of average annual water runoff of rivers of the Siverskyi Donets Basin were identified, cyclic component was identified, duration and nature of water runoff cycles and within them low and wet phases were identified. The prediction of annual water runoff in the near future was done. Joint autocorrelation and spectral analysis allowed to identify a mutually confirmed cycle lasting 19-24 years for the rivers of the Siverskyi Donets Basin. According to the criterion of series, it is established that the periods of low phase can be 9±2 years. Knowing the length of the cycles, water-specific phases were identified. As a result, for the rivers of the Siverskyi Donets Basin it was found that the runoff in the wet phase exceeds the norm by an average of 18%, and the runoff in the low phase is lower by an average of 17% than normal. Therefore, the difference in river water phases is ≈ 35%. Forecast estimates show that by 2020±2, the rivers of the basin will have a low phase, which began in 2008. A wet phase is expected from 2020±2 to 2029±2, then in the period 2030±2 – 2038±2 years – low phase. In the period from 2039±2 to 2048±2, we should expect an increase of annual water runoff of the rivers of the Siverskyi Donets Basin.

Aunque la arqueología de Al-Andalus no se constituyó como disciplina propia hasta hace unos cuarenta años, el interés hacia las fortalezas llamadas por entonces “hispano-musulmanas” se manifestó mucho antes, centrado esencialmente en los aspectos arquitectónicos y de poliorcética. Más recientemente, la aproximación al proceso de la fortificación medieval se ha ido diversificando. No son pocos los trabajos que buscan en él unas respuestas a preguntas más ambiciosas, relativas a la organización de la sociedad campesina, a las estructuras de poblamiento y a la ordenación del territorio, o a las formas adoptadas por el control estatal.Después de unas breves observaciones introductorias sobre el cambio metodológico experimentado a finales de los años 1970, el artículo hace hincapié en la polisemia de los términos árabes referidos a la arquitectura defensiva. A continuación, se centra en las polémicas surgidas a propósito de uno de estos términos, el ḥiṣn (en el ámbito rural) y en las hipótesis avanzadas al respecto. Finalmente, plantea la cuestión de la existencia en al-Andalus de graneros colectivos fortificados y de ribāṭ-s, estructuras mejor documentadas en África del Norte. Palabras claves: fortificación, estructura social islámica, ordenación del espacioTopónimos: al-AndalusPeriodo: siglos VIII-XV ABSTRACTUntil about forty years ago, the archeology of al-Andalus was not regarded as a discipline in itself. However, interest in the so-called “Moorish” fortresses had been expressed much earlier, focused primarily on architectural and polyorcetic aspects. More recently, the approach to the process of medieval fortification has become more diverse: today many scholars seek within it answers to more ambitious questions, related to the organization of peasant society, settlement structures, land-use planning, or formulae of state control.After some preliminary remarks on the methodological change that occurred in the late 1970s, this paper emphasizes the polysemy of Arabic terms referring to defensive architecture. It then focuses on the debates that arose with regard to one of these terms, the ḥiṣn (in rural areas), and on the hypotheses proposed in this respect. Finally, the paper raises the question of the existence in al-Andalus of ribāṭ-s and fortified collective granaries, structures long considered to be specific to North Africa. Keywords: fortification, Islamic social structure, spatial planningPlace names: al-AndalusPeriod: 8th-15th centuries REFERENCIASAcién Almansa, M. 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Abstract This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the $$H^{4}\times H^{3}$$ H 4 × H 3 framework. At first, the lower bound of decay rate for the global solution converging to constant equilibrium state (1, 0) in $$L^2$$ L 2 -norm is $$(1+t)^{-\frac{3}{4}}$$ ( 1 + t ) - 3 4 if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the $$k(k\in [1, 3])$$ k ( k ∈ [ 1 , 3 ] ) th-order spatial derivatives of solution converging to zero in $$L^2$$ L 2 -norm is $$(1+t)^{-\frac{3+2k}{4}}$$ ( 1 + t ) - 3 + 2 k 4 . Finally, it is proved that the lower bound of decay rate for the time derivatives of density and velocity converging to zero in $$L^2$$ L 2 -norm is $$(1+t)^{-\frac{5}{4}}$$ ( 1 + t ) - 5 4 .

AbstractBased on an L1 interpolation operator, a new high-order compact finite volume scheme is derived for the 2D multi-term time fractional sub-diffusion equation. It is shown that the difference scheme is unconditionally convergent and stable in $L_{\infty }$ L ∞ -norm. The convergence order is $O(\tau ^{2-\alpha }+h_{1}^{4}+h_{2}^{4})$ O ( τ 2 − α + h 1 4 + h 2 4 ) , where τ is the temporal step size and $h_{1}$ h 1 is the spatial step size in one direction, $h_{2}$ h 2 is the spatial step size in another direction. Two numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

Abstract This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.

Tujuan dalam penelitian ini adalah untuk mengetahui Peran Komitmen organisasional yang terdiri dari komitment afektif, normative, dan kontinuan dan Kompensasi baik itu kompensasi finansial dan non-finansial terhadap Kepuasan kerja dengan moderasi Budaya organisasi sebagai variabel penguat atau memperlemah pada karyawan PT.PLN (Persero) Unit Induk Wilayah Kalimantan Barat. Sampel dalam penelitian ini adalah 200 orang karyawan dan data yang dapat di olah sebanyak 200 sampel. PT.PLN (Persero) Unit Induk Wilayah Kalimantan Barat. Data dianalisis menggunakan WrapPls 6.0 dan SPSS 16 untuk menguji Uji asumsi Normalitas dan Linieritas.Hasil penelitian ini menyimpulkan bahwa komitmen organisasi berpengaruh positif terhadap kepuasan kerja karyawan PT.PLN (Persero) Unit Induk Wilayah Kalimantan Barat. Kompensasi juag berpengaruh positif terhadap kepuasan kerja karyawan PT.PLN (Persero) Unit Induk Wilayah Kalimantan Barat. 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Re-Framing Participation in the Architecture Studio Recently, within design literature, significant attention has been given to collaboration across different disciplines (see for instance, Nicolini et al.; Carlile), as well as consideration of the breakdown of traditional disciplinarity and the corresponding involvement of users in co-generation (Sanders and Stappers, “Co-Creation” 11–12) through the development and deployment of structured methods and toolkits (Sanders et al., “Framework”; Sanders and Stappers, “Probes”). Relatively less attention has been paid to the workings of the “communities of practice” (Wenger) operating within the disciplinary domain of architecture. The discourse around concept design in architecture has tended to emphasise individualist approaches driven by personal philosophies, inspirations, imitation of a more experienced designer, and emphasis on latent talent or genius (for instance, Moneo). This can be problematic because without a shared language and methods there are limited opportunities for making meaning to facilitate participation between collaborators in architectural studio settings. It is worth asking then: are there things that “Architecture” might learn from “Design” about the deployment of structured methods, and might this interdisciplinary exchange promote participatory practices in studio-based cultures? We address this question by connecting and building on two important concepts relevant to design methods, meta-design as described in the open design literature (De Mul 36–37), and design frames as described by Schön and formalised by Dorst (‘Core’; Frame; see also Weedon). Through this combination, we propose a theory of participation by making shared meaning in architectural design. We animate our theoretical contribution through a design toolkit we have developed, refined, and applied over several years in typologically focused architectural design studios in Australian university contexts. One important contribution, we argue, is to the area of design theory-building, by taking two previously unrelated concepts from the design methods literature. We draw them together using an example from our own design practices to articulate a new term and concept for making shared meaning in design. The other contribution made is to the translation of this concept into the context of studio-based architectural practice, a setting that has traditionally struggled to accept structured methods. The existence of other form-metaphor design tools available for architecture and the theoretical basis of their development and connection to design literature more broadly has not always been clearly articulated (see for example Di Mari and Yoo; Lewis et al.). The rationale for giving an account of the construction and deployment of our own toolkit is to illustrate its theoretical contribution while providing the basis for future field testing and translation (including by other researchers), noting the established trajectory of this kind of work in the design literature (see, for example, Hoolohan and Browne; Visser et al.; Vaajakallio and Mattelmäki; Sanders and Stappers, “Co-Creation” and ”Probes”). In line with this issue’s thematic and epistemological agenda, we adopt what Cross identifies as “designerly ways of knowing” (223), and is at least partly a reflection on a practice in which we engage with our own disciplines and research interests to propose and deploy design thinking as a kind of critical “reflection-in-action” (Robertson and Simonsen 2). Meta-Framing: Combining Meta-Design and Framing Meta-design is a term used in open design literature to describe approaches aimed toward orchestration of a project in such a way that people are afforded the agency to become effective co-designers, regardless of their pre-existing skills or design-specific knowledge (De Mul 36). According to a meta-design approach, design is conceived of as a shared project of mutual learning instead of an individualistic expression of singular genius. Through the establishment of shared protocols and formats, what Ehn (1) calls “infrastructuring”, individuals with even very limited design experience are provided scaffolds that enable them to participate in a design project. One important way in which meta-design helps “create a pathway through a design space” is through the careful selection and adoption of shared guiding metaphors that provide common meanings between co-designers (De Mul 36). The usefulness of metaphors is also recognised in the context of design frames, the second concept on which we build our theory. Conceptualised as “cognitive shortcuts” for making “sense of complex situations” (Haase and Laursen 21), design frames were first conceived of by Schön (132) as a rational approach to design, one guided by “epistemological norms”. Frames have subsequently been further developed within the design methods literature and are defined as a system of counterfactual design decision-making that uses metaphors to provide a rationale for negotiating ill-structured problems. According to Dorst, frames involve: the creation of a (novel) standpoint from which a problematic situation can be tackled … . Although frames are often paraphrased by a simple metaphor, they are in fact very complex sets of statements that include the specific perception of a problem situation, the (implicit) adoption of certain concepts to describe the situation, a ‘working principle’ that underpins a solution and the key thesis: IF we look at the problem situation from this viewpoint, and adopt the working principle associated with that position, THEN we will create the value we are striving for. (525) Despite Schön choosing to illustrate his original conception of framing through the example of a student’s architectural design project, there has been limited subsequent consideration of framing in architectural studio contexts—an exception being Eissa in 2019. This may be because formalised design methods have tended to be treated with suspicion within architectural culture. For instance, Christopher Alexander’s Pattern Language is one such “highly systemised design process” (Dawes and Ostwald 10) that despite its potential to guide participatory design has had an “uneven reception” (Bhatt 716) within architecture itself. One way architecture as a disciplinary domain and as a profession has attempted to engage with design method is through typology, which is one of the few persistent and recurring notions in architectural discourse (Bandini; Grover et al.). As a system of classification, typology categorises “forms and functions as simply and unequivocally as possible” (Oechslin 37). In addition to being used as a classification system, typology has also been positioned as “a process as much as an object”, one with the potential for an “active role in the process of design” (Lathouri 25). Type and typology have been conceptualised as a particular way of projecting architecture’s “disciplinary agency” (Jacoby 936), and this goes some way to explaining their enduring value. A potentially valid criticism of framing is that it can tend toward “design fixation”, when a pre-existing assumption “inadvertently restricts the designers’ imagination” (Crilly). Similarly, typology-as-method—as opposed to a classification tool—has been criticised for being relatively “inflexible” or “reductive” (Shane 2011) and responsible for perpetuating “conservative, static norms” (Jacoby 932) if applied in a rote and non-reflexive way. We deal with these concerns in the discussion of the deployment of our Typekit below. We are drawing here on our experience teaching in the first two years of undergraduate architecture degrees in Australian university settings. As well as being equipped with a diversity of educational, social, linguistic, and cultural backgrounds, students typically have divergent competencies in the domain-specific skills of their discipline and a limited vocabulary for making shared meaning in relation to an architectural proposal. The challenge for studio-based collaborative work in such a context is developing shared understandings and a common language for working on a design project to enable a variety of different design solutions. The brief for a typical studio project will specify a common site, context, and program. Examples we have used include a bathhouse, fire station, archive, civic centre, and lifesaving club. There will then be multiple design solutions proposed by each studio participant. Significantly we are talking about relatively well-structured problems here, typically a specific building program for a specified site and user group. These are quite unlike the open-ended aims of “problem frames” described in the design thinking literature “to handle ill-defined, open-ended, and ambiguous problems that other problem-solving methodologies fail to handle” (Haase and Laursen 21). However, even for well-structured problems, there is still a multitude of possible solutions possible, generated by students working on a particular project brief. This openness reduces the possibility of making shared meaning and thus hinders participation in architectural design. Designing the Typekit The Typekit was developed heuristically out of our experiences teaching together over several years. As part of our own reflective practice, we realised that we had begun to develop a shared language for describing projects including that of students, our own, precedents and canonical works. Often these took the form of a simple formal or functional metaphor such as “the building is a wall”; “the building is an upturned coracle”; or “the building is a cloud”. While these cognitive shortcuts proved useful for our communication there remained the possibility for this language to become esoteric and exclusionary. On the other hand, we recognised the potential for this approach to be shared beyond our immediate “interpretive community” (Fish 485) of two, and we therefore began to develop a meta-design toolkit. Fig. 1: Hybrid page from the Typekit We began by developing a visual catalogue of formal and functional metaphors already present within the panoply of constructed contemporary architectural projects assembled by surveying the popular design media for relevant source material. Fig. 2: Classification of contemporary architectural built work using Typekit metaphors We then used simple line drawings to generate abstract representations of the observed building metaphors adopting isometry to maintain a level of objectivity and a neutral viewing position (Scolari). The drawings themselves were both revelatory and didactic and by applying what Cross calls “designerly ways of knowing” (Cross 223) the toolkit emerged as both design artefact and output of design research. We recognised two fundamentally different kinds of framing metaphors in the set of architectural projects we surveyed, rule-derived and model-derived—terms we are adapting from Choay’s description of “instaurational texts” (8). Rule-derived types describe building forms that navigate the development of a design from a generic to a specific form (Baker 70–71) through a series of discrete “logical operators” (Choay 134). They tend to follow a logic of “begin with x … perform some operation A … perform some operation B … end up with y”. Examples of such operations include add, subtract, scale-translate-rotate, distort and array. Model-derived framing metaphors are different in the way they aim toward an outcome that is an adapted version of an ideal initial form. This involves selecting an existing type and refining it until it suits the required program, site, and context. Examples of the model-derived metaphors we have used include the hedgehog, caterpillar, mountain, cloud, island, and snake as well as architectural Ur-types like the barn, courtyard, tent, treehouse, jetty, and ziggurat. The framing types we included in the Typekit are a combination of rule-derived and model-derived as well as useful hybrids that combined examples from different categories. This classification provides a construct for framing a studio experience while acknowledging that there are other ways of classifying formal types. Fig. 3: Development of isometric drawings of metaphor-frames After we developed a variety of these line drawings, we carried out a synthesis and classification exercise using a version of the KJ method. Like framing, KJ is a technique of abduction developed for dealing objectively with qualitative data without a priori categorisation (Scupin; Kawakita). It has also become an established and widely practiced method within design research (see, for instance, Hanington and Martin 104–5). Themes were developed from the images, and we aimed at balancing a parsimony of typological categories with a saturation of types, that is to capture all observed types/metaphors and to put them in as few buckets as possible. Fig. 4: Synthesis exercise of Typekit metaphors using the KJ method (top); classification detail (bottom) Deploying the Typekit We have successfully deployed the Typekit in architectural design studios at two universities since we started developing it in 2018. As a general process participants adopt a certain metaphor as the starting point of their design. Doing so provides a frame that prefigures other decisions as they move through a concept design process. Once a guiding metaphor is selected, it structures other decision-making by providing a counterfactual logic (Byrne 30). For instance, if a building-as-ramp is chosen as the typology to be deployed this guides a rationale as to where and how it is placed on the site. People should be able to walk on it; it should sit resolutely on the ground and not be floating above it; it should be made of a massive material with windows and doors appearing to be carved out of it; it can have a green occupiable roof; quiet and private spaces should be located at the top away from street noise; active spaces such as a community hall and entry foyer should be located at the bottom of the ramp … and so on. The adoption of the frame of “building-as-ramp” by its very nature is a crucial and critical move in the design process. It is a decision made early in the process that prefigures both “what” and “how” types of questions as the project develops. In the end, the result seems logical even inevitable but there are many other types that could have potentially been explored and these would have posed different kinds of questions and resulted in different kinds of answers during the process. The selection of a guiding metaphor also allows students to engage with historical and contemporary precedents to offer further insights into the development—as well as refinement—of their own projects within that classification. Even given the well-structured nature of the architectural project, precedents provide useful reference points from which to build domain-specific knowledge and benchmarks to measure the differences in approaches still afforded within each typological classification. We believe that our particular meta-framing approach addresses concerns about design fixation and balances mutual learning with opportunities for individual investigation. We position framing less about finding innovative solutions to wicked problems to become more about finding ways for a group of people to reason together through a design problem process by developing and using shared metaphors. Thus our invocation of framing is aligned to what Haase and Laursen term “solution frames” meaning they have an “operational” meaning-making agenda and provide opportunities for developing shared understanding between individuals engaged in a given problem domain (Haase and Laursen 20). By providing a variety of opportunities within an overarching “frame of frames” there are opportunities for parallel design investigation to be undertaken by individual designers. Meta-framing affords opportunities for shared meaning-making and a constructive discourse between different project outcomes. This occurs whether adopting the same type to enable questions including “How is my building-as-snake different from your building-as-snake?”, “Which is the most snake-like?”, or different types (“In what ways is my building-as-ramp different to your building-as-stair?”) By employing everyday visual metaphors, opportunities for “mutual learning between mutual participants” (Robertson and Simonsen 2) are enhanced without the need for substantial domain-specific architectural knowledge at a project’s outset. We argue that the promise of the toolkit and our meta-framing approach more generally is that it actually multiples rather than forecloses opportunities while retaining a shared understanding and language for reasoning through a project domain. This effectively responds to concerns that typology-as-method is a conservative or reductive approach to architectural design. It is important to clarify the role of our toolkit and its relationship to our theory-building agenda. On the basis of the findings accounted for here we do claim to draw specific conclusions about the efficacy of our toolkit. We simply did not collect experimental data relevant to that task. We can, however, use the example of our toolkit to animate, flesh out, and operationalise a model for collaboration in architectural design that may be useful for teaching and practicing architecture in collaborative, team-based contexts. The contribution of this account, therefore, is theoretical. That is, the adaptation of concepts from design literature modified and translated into a new domain to serve new purposes. The Promise of Meta-Framing through Typology Through our work, we have outlined the benefits of adopting formalised design methods in architecture as a way of supporting participation, including using toolkits for scaffolding architectural concept design. Meta-framing has shown itself to be a useful approach to enable participation in architectural design in a number of ways. 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